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Departamento de Física Teórica, UV – IFIC (CSIC)

Neutrino masses and dark matter:a path to new physics

PhD thesis by

Juan Andrés Herrero García

Under the supervision of

Nuria Rius Dionis

Arcadi Santamaria i Luna

Valencia, June 2014

NURIA RIUS DIONIS, profesora titular del Departamento de FísicaTeórica de la Universidad de Valencia, y

ARCADI SANTAMARIA I LUNA, catedrático del Departamento deFísica Teórica de la Universidad de Valencia,

Certifican:

Que la presente memoria, “NEUTRINO MASSES AND DARK MAT-TER: A PATH TO NEW PHYSICS” ha sido realizada bajo su dirección en elInstituto de Física Corpuscular, centro mixto de la Universidad de Valencia ydel CSIC, por JUAN ANDRÉS HERRERO GARCÍA, y constituye su Tesispara optar al grado de Doctor en Ciencias Físicas.

Y para que así conste, en cumplimiento de la legislación vigente, pre-senta en el Departamento de Física Teórica de la Universidad de Valencia lareferida Tesis Doctoral, y firman el presente certificado, en

Valencia, 10 de Mayo de 2014

Nuria Rius DionisArcadi Santamaria i Luna

A mis padres

If you can dream - and not make dreams your master...If you can meet with Triumph and Disaster

and treat those two impostors just the same...If you can talk with crowds and keep your virtue,

or walk with kings - nor lose the common touch...Yours is the Earth and everything that is in it,

And -which is more- you will be a Man my son!Rudyard Kipling

Every man dies, not every man really lives.William Wallace - Braveheart

El cerebro no está para descubrir la verdad, sino para hacerpredicciones para poder sobrevivir.

Richard Gregory, profesor emérito de neuropsicología dela U. Bristol.

El destino es el que baraja las cartas, pero somos nosotros los quejugamos.

Shakespeare

Nada más cumplir siete tuve que interrumpir mi educación parair a la escuela.

Bernard Shaw

Es mejor tener la boca cerrada y parecer estúpido que abrirla ydisipar la duda.

Mark Twain

La edad de piedra no se acabó porque se acabaran las piedras.Jeque Yamani

Es más fácil desintegrar un átomo que un prejuicio.No entiendes realmente algo a menos que seas capaz de

explicárselo a tu abuela.Albert Einstein

Jamás diga una mentira que no pueda probar.Millor Fernandes

Sólo creo en las estadísticas que yo mismo falsifico.Winston Churchill

Robar un banco no es nada comparado con fundarlo.Bertolt Brecht

Gasté mucho dinero en licor, mujeres, juergas y cochesdeportivos. El resto, lo desperdicié.

George Best

En aprender cómo se juega al póker se tarda un minuto, encontrolar los aspectos del juego toda una vida.

Mike Sexton

Y ganar, ganar, ganar, y volver a ganar, ganar ganar, y volver aganar, ganar, ganar...

Luis Aragonés

List of publications

This PhD thesis is based on the following publications:∙ On the nature of the fourth generation neutrino and its implications [1]

A. Aparici, J. Herrero-Garcia, N. Rius and A. Santamaria.JHEP 1207 (2012) 030

∙ Neutrino masses from new generations [2]A. Aparici, J. Herrero-Garcia, N. Rius and A. Santamaria.JHEP 1107 (2011) 122

∙ The Zee-Babu model revisited in light of new data [3]J. Herrero-Garcia, M. Nebot, N. Rius and A. Santamaria.sent to NPB

∙ On the annual modulation signal in dark matter direct detection [4]J. Herrero-Garcia, T. Schwetz and J. Zupan.JCAP 1203 (2012) 005

∙ Astrophysics independent bounds on the annual modulation of darkmatter signals’ [5]J. Herrero-Garcia, T. Schwetz and J. Zupan.Phys.Rev.Lett. 109 141301

∙ Halo-independent methods for inelastic dark matter scattering [6]N. Bozorgnia, J. Herrero-Garcia, T. Schwetz and J. Zupan.JCAP 1307 (2013) 049

and the following contributions to proceedings:∙ Can New Generations Explain Neutrino Masses? [7]

A. Aparici, J. Herrero-Garcia, N. Rius and A. Santamaria.Moriond proceedings

∙ Implications of new generations on neutrino masses [8]A. Aparici, J. Herrero-Garcia, N. Rius and A. Santamaria.J.Phys.Conf.Ser. 408 012030 (2013)

http://arxiv.org/abs/arXiv:1204.1021








ii List of publications

Other works not included in this thesis:

∙ Neutrino masses, Grand Unification, and Baryon Number Violation [9]A. de Gouvea, J. Herrero-Garcia and A. Kobach.sent to PRD

http://arXiv.org/abs/arXiv:1404.4057

Abbreviations

The following abbreviations have been used throughout the text:

SM Standard Model

BSM beyond the Standard Model

HEP high energy physics

SSB spontaneous symmetry breaking

VEV vacuum expectation value

EFT effective field theory

EWS electroweak scale

EWSB electroweak symmetry breaking

LFV lepton flavour violation

LN lepton number

LNV lepton number violation

NP new physics

QCD Quantum Chromodynamics

QED Quantum Electrodynamics

QFT Quantum Field Theory

0𝜈𝛽𝛽 neutrinoless double beta decay

CL confidence level

LHC Large Hadron Collider

iv Abbreviations

PMNS Pontecorvo-Maki-Nagakawa-Sakata (or lepton mixing) matrix

NH Normal Hierarchy

IH Inverted Hierarchy

QD quasi-degenerate spectrum

CKM Cabibbo-Kobayashi-Maskawa (or quark mixing) matrix

DM dark matter

CDM cold dark matter

WDM warm dark matter

DE dark energy

ΛCDM Cosmological Model (Λ: cosmological constant, CDM: cold dark mat-ter)

FLRW metric Friedmann-Lemaitre-Robertson-Walker metric

GR General Relativity

RD radiation-domination

MD matter-domination

MW Milky Way

MOND Modified Newtonian Dynamics

f.o. freeze-out

WIMP Weakly Interacting Massive Particle

SHM Standard Halo Model

SI spin-independent

SD spin-dependent

IV isospin-violating

SHM Standard Halo Model

Resumen

Resumimos aquí en castellano la investigación realizada en esta tesis doctoral,las principales conclusiones obtenidas y la estructura de la misma.

Esta tesis doctoral se ha centrado en dos de las evidencias más convincentespara la necesidad de física más allá del Modelo Estándar (SM): las masasde los neutrinos y la materia oscura (DM). La escala de esta nueva física(NP) es completamente desconocida, por lo que hemos adoptado un enfoquefenomenológico, con su testabilidad como un objetivo primordial. En estesentido, nos hemos centrado en los modelos radiativos de masas de neutrinos,a la escala del TeV, y se ha supuesto que la materia oscura es una partículamasiva de interacción débil (WIMP), de masa 𝒪(1 − 103) GeV.

La primera parte de la tesis se dedica al estudio de modelos de masasde neutrinos generadas radiativamente. Éstos explican por qué las masasde los neutrinos son mucho menores que las del resto de los fermiones: losneutrinos no tienen masa a nivel árbol, siendo su masa generada radiativa-mente a uno, dos o tres loops (más loops típicamente producen masas de losneutrinos demasiado pequeñas y/o tienen problemas con las restricciones quevienen de no observar procesos que violen sabor (LFV) en leptones cargados).Además, gracias a la supresión de los loops y a la presencia obligatoria devarios acoplamientos para violar número leptónico (LN), estos modelos tienenpartículas a una escala lo suficientemente baja como para dar señales encolisionadores, como el LHC, y en los experimentos de baja energía como losque buscan 𝜇 → 𝑒𝛾 o 0𝜈𝛽𝛽.

En esta tesis se ha estudiado la generación de la masa de los neutrinosradiativamente a dos loops, tanto a través de nuevos fermiones (nuevasfamilias) como de nuevos escalares (el modelo Zee-Babu). También se estudióla conexión de estos modelos de masas de neutrinos con otras ramas de físicade altas energías, como la física del Higgs, ya sea indirectamente, por ejemploa través de modificaciones de la señal de 𝐻 → 𝛾𝛾, o directamente, a travésde la detección de las nuevas partículas.

En los primeros trabajos, se estudió en detalle la posibilidad de la existenciade una cuarta generación secuencial del SM, abordando en particular las

vi Resumen

propiedades de los neutrinos ligeros y su naturaleza [1]. El neutrino de lacuarta generación debe ser mucho más pesado que los otros, por lo que laopción más plausible es la existencia de un neutrino dextrógiro para darlemasa. Nos dimos cuenta de que si los neutrinos ligeros son partículas deMajorana, hay una contribución a la masa de Majorana de este neutrinodextrógiro a dos loops. Por lo tanto, el neutrino de la cuarta generacióntambién debe ser Majorana, a menos que se imponga cierta simetría en losleptones de la cuarta generación.

Se analizaron en detalle las posibles implicaciones de tener un neutrinopesado de Majorana de la cuarta generación. En particular, se generanmasas de los neutrinos ligeros a dos loops, y esta contribución puede superarfácilmente el límite cosmológico de la escala de masa absoluta de los neutrinos.

Sin embargo, incluso si hay una contribución a las masas de los neutrinosligeros, su espectro no se puede explicar con una única familia adicional. Porlo tanto, en un trabajo diferente, se estudió la posibilidad de que los neutrinosligeros fueran sin masa a nivel árbol, con sus masas generadas a dos loopspor la acción de los nuevos fermiones, por ejemplo, con dos generacionesadicionales [2].

Actualmente, gracias al LHC, la existencia de nuevas generaciones secuen-ciales de partículas ha sido descartada, al menos si el sector escalar comprendesólo el bosón de Higgs del SM, debido al hecho de que la producción delbosón de Higgs aumenta con la presencia de los nuevos fermiones (y tambiénsus desintegraciones varían significativamente si existe una nueva familia).Existen algunas posibilidades que pueden salvar a la cuarta generación, comoun sector escalar mayor, por ejemplo, con un doblete de Higgs adicional quesólo se acopla a la cuarta familia [10]. Aun así, cotas a los fermiones de lacuarta generación están muy cerca del límite perturbativo: en el caso de losquarks & 600 GeV, véase, por ejemplo, [11].

En otro trabajo hemos revisado el modelo de Zee-Babu [3] a la luz de losnuevos datos, por ejemplo, el ángulo de mezcla 𝜃13, el nuevo bound de 𝜇 → 𝑒𝛾y los resultados del LHC. También se analiza la posibilidad de explicar lasdesviaciones respecto al SM en 𝐻 → 𝛾𝛾 vista por ATLAS (aunque no es unadesviación significativa).

Nos encontramos con que los datos de oscilación de neutrinos y las re-stricciones de bajas energías siguen siendo compatibles con masas de losnuevos escalares accesibles al LHC. Si alguno de los escalares es descubierto,el modelo puede ser falsable por la combinación de información sobre losmodos de desintegración con los datos de neutrinos. Por el contrario, si seencuentra que el espectro de neutrinos es invertido y la fase CP 𝛿 es muydiferente de ∼ 𝜋, las masas de los escalares cargados estarán fuera del alcancedel LHC.

Resumen vii

En un sentido amplio, se ha tratado de arrojar algo de luz sobre las difer-entes formas de generar masas de los neutrinos radiativamente, centrándonosen que fueran comprobables, siempre con la mirada puesta en las posiblesseñales en el LHC, y en su conexión con otras ramas de la física de altasenergías. Esperemos que, con más información experimental, la naturalezanos ayude a averiguar de qué forma entre la larga lista de posibilidadesganan masa los neutrinos. Por el momento, podemos decir que las familiasadicionales se encuentran en muy mal estado, mientras que el modelo deZee-Babu sigue vivo y listo para ser descubierto en el LHC!

Respecto a la DM, nos hemos dedicado al estudio de los límites que puedenponerse en su modulación anual. Si la DM son partículas que interaccionandébilmente con el SM (WIMPs), pueden ser observadas en experimentos dedetección directa. Éstos son muy difíciles de realizar, ya que el número deeventos esperado de la DM es muy bajo y las señales del fondo son muygrandes, incluso cuando los experimentos se colocan en laboratorios a muchaprofundidad y debidamente blindados.

La modulación anual que debe estar presente en la señal de experimentosde detección directa, debido al movimiento relativo de la Tierra alrededor delSol, podría ayudar a discriminar una señal de DM del fondo (algunas señalesde fondo modulan, como los muones, pero la fase no tiene por qué coincidircon la esperada de DM, ver capítulo 6).

Un punto importante a tener en cuenta en la interpretación de una señalDM es que las tasas de eventos están sujetas a incertidumbres astrofísicas,como la distribución de velocidades de la DM, la densidad local o la velocidadde escape, por lo que es importante ser lo más independiente de los parámetrosastrofísicos como sea posible. Ésto se puede hacer representando la integralde la distribución de velocidades dividida entre la velocidad (𝜂(𝑣𝑚, 𝑡), dónde𝑣𝑚 es la mínima velocidad necesaria para producir un retroceso nuclear deuna energía dada, véase eq. 6.10) de los diferentes experimentos en el mismorango de velocidades (𝑣𝑚).

En esta tesis, hemos deducido restricciones a la modulación anual entérminos de la tasa total de eventos (no modulada), que son casi independientesde las incertidumbres astrofísicas, mediante el desarrollo de 𝜂(𝑣𝑚, 𝑡) a primerorden en la velocidad, 𝑣𝑒(𝑡)/𝑣, donde 𝑣𝑒(𝑡) es la velocidad de la Tierra alrededordel Sol y 𝑣 es la velocidad de la DM. Este test es una prueba importante quecualquier señal de modulación anual tiene que pasar.

Aplicamos estos límites a las modulaciones vistas por DAMA y CoGeNT [4],y obtuvimos que la modulación de DAMA es compatible con su tasa constante,mientras que la de CoGeNT está excluida para halos típicos de DM a & 90%CL.

En un segundo trabajo demostramos como las cotas de la modulación

viii Resumen

de un experimento en función de su tasa constante, obtenidas en el anteriortrabajo, pueden aplicarse entre experimentos diferentes, en el mismo rango develocidades (𝑣𝑚). Los aplicamos a la modulación de DAMA frente a los resul-tados nulos de XENON, CDMS y otros experimentos [5]. Una interpretaciónen términos de DM de la modulación de DAMA está desfavorecida por almenos un experimento en más de 4𝜎, para una masa de la DM . 15 GeV,para todo tipo de interacciones.

También hemos ampliado el análisis al caso de dispersión inelástica [6],mostrando como DAMA es incompatible con XENON 100 también en estecaso.

En un sentido amplio, hemos hallado límites independientes de la astrofísicaque las modulaciones anuales tienen que cumplir. Con suerte, este tipo depruebas va a ayudar a discriminar entre señales verdaderas de DM y fondos. Esimportante destacar que es una condición necesaria para que la modulación seade DM, pero no suficiente. Después de aplicar los límites a las modulacionesanuales de DAMA y CoGeNT y compararlos con los otros experimentos queno ven DM, éstas están muy desfavorecidas interpretadas como debidas aDM. Esperamos que en los próximos años más datos experimentales ayudena clarificar la actual situación, que es confusa, y esperemos que la DM estéallí fuera lista para ser descubierta de una manera u otra!

La tesis (ya en inglés, el lenguaje científico por excelencia) está estructuradade la siguiente forma. Comienza con la motivación de la necesidad de nuevafísica (NP) en la parte I. Como los dos temas principales de esta tesis, lasmasas de los neutrinos y la materia oscura, aunque son dos de las formas másinteresantes para buscar NP, son muy diferentes en la naturaleza1, hemosdividido la tesis en dos partes independientes.

En la parte II damos una introducción al tema de la masa de los neutrinos.Después de una breve introducción al SM en el capítulo 1, y al uso deteorías efectivas (EFT) con el Operador Weinberg en mente, se estudia lafenomenología de neutrinos en el capítulo 2, dando una introducción a laviolación de número leptónico (LNV). Antes de proponer cualquier cosa, unotiene que saber lo que hay ya propuesto, y el capítulo 3 intenta realizaresta tarea, con una revisión de algunos de los modelos más populares de laliteratura.

La parte III es una introducción al tema de la materia oscura (DM). Trasuna corta descripción del modelo cosmológico en el capítulo 4, motivamosla necesidad de materia oscura, enumerando sus propiedades y revisandobrevemente la posibilidad de detectarla de forma indirecta y en colisionadores

1Al menos en principio, aunque puede haber conexiones entre ellos, véase el modelo deldoblete inerte en el capítulo 3, sección 3.6.2.

Resumen ix

en el capítulo 5. En el capítulo 6 nos centramos en la detección directa de laDM, en su modulación anual, presentando un análisis de la distribución develocidades y una forma de presentar los datos que es independiente de laastrofísica, y terminando con un breve resumen de la situación experimentalen detección directa de DM.

En la parte IV, hacemos un resumen de los resultados obtenidos y algunasobservaciones finales sobre la investigación realizada. Por último, en la parteV presentamos una copia de los artículos de investigación realizados para estatesis, tal y como se publican en las diferentes revistas. Estos artículos resumenla principal labor de investigación realizada estos años, y la discusión, lainterpretación de los resultados y las conclusiones de cada uno de los trabajospresentados son de importancia, siendo lo presentado aquí (y en la parte IV)únicamente un conciso resumen de las mismas.

Preface

In this doctoral thesis we have focused on two of the most compelling evidencesfor physics beyond the Standard Model (SM): neutrino masses and darkmatter (DM). The new physics scale regarding these two topics is completelyunknown, so we have adopted a phenomenological approach, with testabilityas a prime goal. In this sense, we have focused on TeV scale neutrino massmodels (radiative) and we have assumed that DM is a Weakly InteractingMassive Particle, WIMP, with mass 𝒪(1 − 103) GeV.

We start the thesis by motivating the need for new physics (NP) in part I.As the two main topics of this thesis, neutrino masses and DM, while beingtwo of the more interesting ways to look for NP, are quite different in nature2,we have divided the thesis into two independent parts.

In part II we give an introduction to the topic of neutrino masses. Aftera brief introduction to the SM in chapter 1, and the use of effective theories(EFT) with the Weinberg Operator in mind, we study the phenomenology ofneutrinos in chapter 2, giving an introduction to the topic of lepton numberviolation (LNV). Before proposing a new model, one needs to know what hasalready been studied in the literature, and chapter 3 tries to accomplish thistask, with a review of some of the most popular models.

In part III we give an introduction to the topic of DM. After a briefintroduction to the Cosmological Model in chapter 4, we motivate the needfor DM, enumerating its properties and reviewing briefly the possibility ofdetecting it indirectly and in colliders in chapter 5. In chapter 6 we focus ondirect detection of DM, introducing the concepts of event rate and annualmodulation signal, and discussing the velocity distribution and how data canbe presented in an astrophysics-independent way. We finish by discussing theexperimental results of DM direct detection.

Regarding the research done on neutrino masses, we have analysed radia-tive neutrino mass models with extra fermions/scalars (in the context of newgenerations and the Zee-Babu model), looking for connections with other NP

2At least in principle, although there are several proposed connections and interplaysbetween them. As a nice example, see the inert doublet model in chapter 3, section 3.6.2.

xii Preface

scenarios. With respect to DM, we have focused on one of the fundamentalways to discover it, which is its direct detection, and, in particular, we havestudied the annual modulation signal and bounds that can be placed on it.In part IV, we give a summary of the results obtained and some concludingremarks on the research done.

Finally, in part V we present a copy of the research articles done for thisthesis, as published in the different journals. This is the main research done inthis thesis, and the discussion, interpretation of the results and the conclusionsof each of the works presented are of relevance, with the conclusions presentedin IV being a concise summary of them.

Contents

List of publications i

Abbreviations iii

Resumen de la tesis v

Preface xi

I Introduction: new physics is needed 1

II Neutrino masses 9

1 The Standard Model in a nutshell 111.1 The gauge group and the fermion content . . . . . . . . . . . . 111.2 The Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . 131.3 Flavour: masses and mixings . . . . . . . . . . . . . . . . . . . 151.4 Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . 17

2 Neutrino masses 192.1 Brief history of neutrino oscillations . . . . . . . . . . . . . . . 192.2 Neutrino parameters . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Unsolved questions in the neutrino sector . . . . . . . . 222.3 Dirac and Majorana masses . . . . . . . . . . . . . . . . . . . 232.4 The Weinberg operator . . . . . . . . . . . . . . . . . . . . . . 25

3 Neutrino mass models 273.1 Type I: fermionic singlets . . . . . . . . . . . . . . . . . . . . . 273.2 Type II: scalar triplet . . . . . . . . . . . . . . . . . . . . . . . 283.3 Type III: fermionic triplets . . . . . . . . . . . . . . . . . . . . 293.4 Inverse seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Contents

3.5 High-energy frameworks . . . . . . . . . . . . . . . . . . . . . 313.5.1 Left-right symmetric models . . . . . . . . . . . . . . . 313.5.2 Supersymmetric models . . . . . . . . . . . . . . . . . 323.5.3 Grand Unified Theories . . . . . . . . . . . . . . . . . . 32

3.6 Radiative models . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.1 The Zee Model . . . . . . . . . . . . . . . . . . . . . . 333.6.2 The Scotogenic model . . . . . . . . . . . . . . . . . . 34

III Dark matter 37

4 The Cosmological Model in a nutshell 394.1 The basic ingredients of the ΛCDM model . . . . . . . . . . . 394.2 Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . 414.3 Hubble’s Law and other cosmological parameters . . . . . . . 424.4 The energy content of the Universe . . . . . . . . . . . . . . . 444.5 An inflationary epoch . . . . . . . . . . . . . . . . . . . . . . . 48

5 Dark matter 535.1 Evidence of dark matter . . . . . . . . . . . . . . . . . . . . . 535.2 Properties of dark matter . . . . . . . . . . . . . . . . . . . . 565.3 Thermal freeze-out and the WIMP paradigm . . . . . . . . . . 595.4 Asymmetric dark matter . . . . . . . . . . . . . . . . . . . . . 615.5 Dark matter searches . . . . . . . . . . . . . . . . . . . . . . . 61

5.5.1 Dark matter indirect detection . . . . . . . . . . . . . . 615.5.2 Collider searches . . . . . . . . . . . . . . . . . . . . . 635.5.3 Imprints in BBN and in the CMB . . . . . . . . . . . . 63

6 Direct detection of dark matter 656.1 The event rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 The annual modulation signal . . . . . . . . . . . . . . . . . . 686.3 The velocity distribution and the 𝜂-𝑣min plot . . . . . . . . . . 706.4 Current experimental situation . . . . . . . . . . . . . . . . . 75

IV Final remarks and conclusions 79

Bibliography 85

Contents

V Scientific Research 103

Philosophical reflection i

Agradecimientos v

Algunas críticas y sugerencias xv

Part I

Introduction: new physics isneeded

Introduction: new physics is needed 3

From ∼ 10−10 s up to today, at ∼ 1017 s, (or equivalently in energy,from the ∼TeV to the current temperature, given by the Cosmic MicrowaveBackground (CMB), ∼ 10−4 eV) the history of the Universe is based on wellknown laws of physics3. Before ∼ 10−10 s, or equivalently above the TeV, upto the Planck scale at ∼ 1016 TeV, almost nothing is known.4

At low scales (high energies, but below the electroweak scale (EWS),∼ 𝒪(100) GeV), the Standard Model (SM) of particle physics, which wewill review in chapter 1, section 1.1, describes almost all known particlephysics processes, and has been tested with great accuracy both in the quarkand in the lepton sectors. In addition, with the discovery of the Higgsboson [13, 14], its missing piece responsible for the breaking of electroweaksymmetry (EWSB), it seems a complete theory.

At large scales (low energies), the ΛCDM Model, based on General Rela-tivity and on the Cosmological Principle, that we will review in chapter 4,section 4.1, is a very successful framework that describes the evolution of theUniverse. Large scale observations, like the Cosmic Microwave Background(CMB), or the abundances of light elements (Nucleosynthesis), are extremelystrong evidences of a Hot Big Bang.

However, the truth is that we have both in particle physics and in cos-mology many unsolved questions. The SM is far from describing all themicro-physics we know of, and we think of it as just a low energy effectivetheory (EFT) valid at energies below the scale of new physics (NP) Λ wherea more fundamental description will kick in (the use of EFT will be reviewedin section 1.4 of chapter 1). In a similar way, ΛCDM describes the dynamicsof the Universe with a few parameters, but unfortunately in many cases itis just a parametrization of the new physics (NP) by just giving it a nameand some very basic properties, like the cosmological constant (Λ) or colddark matter (CDM), but we are far from knowing which new particles andinteractions describe this NP.

As the author thinks that it is of uttermost importance to examine deeplywhy we study something, let us devote some time to analyse why we thinkNP is needed, explaining in more detail the drawbacks of the SM and theΛCDM model, before focusing on the main topics of this thesis. The mainreasons coming from experimental observations are the following:

∙ Dark energy (DE). DE is responsible for the accelerated expansion of3Although some like dark energy and dark matter are still mysteries.4With the discovery of cosmological B-modes made by BICEP2 [12], which are the

imprint of the gravitational waves of inflation, and that should be confirmed by Planck, weare in fact reaching much earlier times in our knowledge of the History of the Universe,times as early as ∼ 10−36 s, or energies of ∼ 1016 GeV.

4 Introduction: new physics is needed

the Universe, as observed in 1998 using Type Ia Supernovae [15, 16].As will be shown in chapter 4, different observations have revealedthat only about ∼ 5% of the Universe is described by the SM, therest being dark matter (∼ 26%) and DE (∼ 69%), for which we don’thave yet an accepted fundamental theory. One option is that it is justa cosmological constant (see [17] for a review), a fluid with negativepressure, with equation of state 𝑝 = −𝜌. However, the value of Λ ismuch smaller than the one expected in QFT, a possible reason beingsome unknown symmetry. It could also happen that DE is due to somedynamical mechanism [18, 19], for instance with a quintessence fieldwith time-dependent 𝑤, or a modified theory of gravity [20].

∙ Dark matter (DM). There is compelling evidence of the existence ofnon-luminous non-baryonic matter in the Universe. A substantial partof this thesis will be devoted to DM and its direct detection. We leaveto chapters 5 and 6 an overview of these topics.

∙ The baryon asymmetry of the Universe (BAU). In the Universe there aremore baryons (b) than anti-baryons (��); annihilations have not erasedall particles, so fortunately matter is present, with a small baryonto photon ratio, 𝜂 = 𝑛𝑏−𝑛��

𝑛𝛾∼ 10−10. This BAU should be generated

dynamically, because even if there was a preexisting asymmetry, fine-tuned to the present value, it would have been diluted after inflation.Several mechanisms exist, like leptogenesis [21], which for instancecan occur via decays of heavy right-handed neutrinos (with masseslarger than 109 GeV [22]) that violate lepton number (LN), and thisasymmetry can be transferred to the baryons via sphaleron processes,which preserve 𝐵 − 𝐿. Note that in general three conditions [23] mustbe satisfied to generate the BAU: violation of 𝐵, violation of C andCP (otherwise processes involving �� would generate the same BAU butopposite in sign as the one generated by 𝑏) and departure from thermalequilibrium. The SM has all the ingredients, however the amount ofCP violation is too small and the transition is not strongly first order.

∙ Neutrino masses. In the SM (defined without right-handed neutrinos),neutrinos are massless. However, we know that neutrinos oscillate andare therefore massive particles that mix. A substantial part of thisthesis will be devoted to neutrino masses and models, so we refer tochapters 2 and 3 for an overview.

From theoretical arguments, we have that the SM is not satisfactory inthe following points:


∙ The so-called flavour puzzle. There is no explanation for the numberof families5, the masses or the mixings of the leptons and quarks. Inparticular, there is a hierarchy in masses between particles of the differentfamilies, which is not understood at all. Many free parameters (∼ 20)are needed to explain all the low-energy data and at the moment nocompelling theory of flavour is able to explain why they take the valueswe have measured. In addition, indirect constraints on flavour changingneutral currents (FCNC) and other LFV processes have pushed the NPscale to be in most cases far away from the EWS.

∙ The strong CP problem. A term in the Lagrangian 𝜃𝜖𝜇𝜈𝜌𝜎𝐺𝜇𝜈𝐺𝜌𝜎,which is a relevant operator, is perfectly allowed by the SM symmetryand violates T and CP. However, bounds from neutron electric dipolemoments yield 𝜃 . 10−9, and the common lore is that Lagrangianparameters should be 𝒪(1) unless a symmetry protects them (this isknown as naturality). A solution is to promote 𝜃 to a field, the axion,charged under the Peccei-Quinn symmetry [24–27]. In addition, it canconstitute the DM of the Universe, with a mass around ∼ 10−5 eV.

∙ The Hierarchy problem. It is not a problem of the SM, but a problemthat the SM has in the presence of NP at a high scale Λ. The Higgsmass is not protected by any symmetry (unlike fermions, protected bychiral symmetry, or gauge bosons, whose mass is protected by the gaugesymmetry) and receives quadratic corrections, ∝ Λ2, which yield thepresence of a weak-scale Higgs boson (𝑚ℎ ∼ 125 GeV) unnatural.6 Inprinciple, this problem could be solved if, like the pions in QCD, theHiggs boson is no longer an elementary scalar particle but a condensateof fermions, like in Technicolor models (see [28] for a classic review).Another possibility is Supersymmetry (see [29–31] for some reviews),where fermions and bosons are related to each other. One shouldalways keep in mind that, if no high energy NP exists, as stated at thebeginning, there is no Hierarchy Problem. However, gravity is alwaysthere, although we do not know how to construct a renormalizablequantum field theory (QFT) of it.

∙ Unification of interactions. The SM group is the direct product of threegauge groups. It would be desirable if these were unified in a unique

5In this thesis we present some works regarding possibility of the existence of a fourthgeneration. The new experimental evidence with the discovery of the Higgs boson stronglydisfavours the presence of extra families.

6Another way to pose the problem is that a large amount of cancellations between thebare mass and the loop corrections is needed, i.e., a high degree of fine-tuning is necessary.

6 Introduction: new physics is needed

group at a higher scale.7 In fact, anomaly cancellation within eachfamily makes clear that leptons know about quarks and viceversa, sothe fact that at some point they are unified in the same representationshould not surprise anyone. In addition, we know that couplings runwith energy, and in the SM, the gauge couplings point to a high-scalearound 1015 GeV where they would unify. However, in the SM theydon’t perfectly unify, so NP between the EW and the GUT scales isneeded to help them converge.8

∙ Charge quantization. It is not understood why protons (𝑞𝑝 = +𝑒, with𝑒 the magnitude of the electron charge) and electrons (𝑞𝑒 = −𝑒) haveopposite electric charges. At a more fundamental level, quarks comein either +2/3 𝑒 or −1/3 𝑒. There is no explanation for this in the SM,while there is in some GUTs, like SU(5) [33].

∙ Gravity. We describe micro-physics with QFT. However, when one triesto quantize gravity, we are dealt with a non-renormalizable theory. Thismakes the unification of gravity with the other SM interactions oneof the most important theoretical problem (if not the most). Therehave been attempts to go beyond QFT, such as String Theory, withhowever limited predictive power. An important fact is to note thatgravity involves the presence of a new energy scale, the Planck scale(𝑚𝑝 =

√~ 𝑐/𝐺 ≈ 1019 GeV), and therefore the Hierarchy problem seems

a real concern, unless the NP makes the existence of very different scalesnatural.

∙ Inflation. In the Standard Cosmological Model ΛCDM, with a Hot BigBang, which will be reviewed in chapter 4, there are some problems(the horizon problem, the flatness problem, the problem of structureformation, or the absence of monopoles and domain walls hypothesizedin some SM extensions...) which are solved by having a period of ac-celerated expansion in the very early Universe (𝑡 ∼ 10−34 s), calledinflation. After the discovery of primordial B-modes caused by primor-dial gravitational waves, made by BICEP2 [12], and which should beconfirmed independently by other experiments, like Planck, the SMneeds to be extended to accommodate inflation. In the literature thereare many testable inflationary models and more data should shed light

7These theories are called Grand Unified Theories (GUTs), and SO(10) is a very niceexample, whose irreducible representation is the 16 and contains the 𝜈𝑅 as a fundamentalcomponent. Lower bounds to the GUT scale exist from proton decay [32].

8In Supersymmetry, depending on the scale of SUSY breaking, unification can beachieved (typically it should kick in below 𝒪(10) TeV).


on which is the one chosen by Nature. The interested reader is referredto [34] for a nice review on inflation.

From all these reasons, we know that NP, possibly in many different scalesΛNP and forms, is needed:

∙ If ΛNP ∼ 𝒪(1) TeV, it may be tested at the Large Hadron Collider(LHC) in its second run at 13 − 14 TeV, but this does not need to bethe case.

∙ If ΛNP ≪ EWS (like the axion, that solves the strong CP problem andis a good DM candidate), the SM would not even be a good EFT, andit may be tested in other type of experiments.

∙ If ΛNP ≫ 𝒪(1) TeV, we may never be able to detect the new particlesdirectly, and we may just have to gather indirect evidence via effects ofthem in other low-energy processes, and by falsifying other possibilities.9However, as we have seen, there are naturality arguments to expect thatif the NP is roughly above 𝒪(10) TeV, some mechanism is needed tosolve the Hierarchy Problem.

In this thesis we focus on NP scenarios with testable scales, at ∼ 𝒪(1)TeV, that try to shed some light on neutrino masses and dark matter.

9This may be the case for SO(10) well-motivated seesaw type I, to be studied in chapter3, section 3.1, which seems a very reasonable explanation to neutrino masses, but verydifficult to test.

Part II

Neutrino masses

Chapter 1

The Standard Model in anutshell

The Standard Model (SM) describes our current knowledge of particle physics.It is a quantum field theory (QFT) that successfully describes the electro-magnetic, the weak and the strong interactions [35–46]. However, it does notdescribe all known phenomena and we know that new physics (NP) is needed,as has been argued in the introduction, part I. In this chapter we will brieflyreview the basic components of the SM.

1.1 The gauge group and the fermion content

The SM is based on the gauge group SU(3)c × SU(2)L × U(1)Y, with themessengers of the interactions being the gauge bosons associated with thegenerators of each group:

∙ 8 gluons 𝐺𝜇 for the strong interactions based on SU(3)c, QuantumChromodynamics (QCD), with 𝑔𝑠 being the strong coupling.

∙ The charged 𝑊 𝜇, the neutral 𝑍𝜇 and the photon 𝐴𝜇 for the unifiedelectroweak interactions, based on the product group SU(2)L × U(1)Y,with 𝑔 being the coupling of the SU(2)L interactions and 𝑔 ′ the couplingof the U(1)Y ones.1

1After spontaneous symmetry breaking (SSB) the third real gauge boson 𝑊 3𝜇 of SU(2)

and the hypercharge one of U(1)Y, 𝐵𝜇, mix to yield the 𝑍 and the photon, the force carrierof Quantum Electrodynamics (QED).

12 Chapter 1. The Standard Model in a nutshell

ℓ =(𝜈𝐿𝑒𝐿

)𝑒R 𝜈R? 𝑄 =

(𝑢𝐿𝑑𝐿

)𝑢R 𝑑R 𝜑

SU(3)c 1 1 1 3 3 3 1

SU(2)L 2 1 1 2 1 1 2

U(1)Y −1/2 −1 0 +1/6 +2/3 −1/3 +1/2

Table 1.1: The Standard Model particle content of one family (quarksand leptons, 𝜓𝛼 = (𝑄, 𝑑R, 𝑢R, ℓ, 𝑒R)) and its representation underSU(3)c × SU(2)L × U(1)Y. Also shown the Higgs field 𝜑. The existenceof right-handed neutrinos 𝜈R is at this moment completely hypothetical, onlybased on simplicity, aesthetic and theoretical reasons.

The covariant derivatives take into account the transformation propertiesof the fields under the SM gauge group:

𝐷𝜇 = 𝜕𝜇 − 𝑖𝑔𝑠𝜆𝑎2 𝐺

𝑎𝜇 − 𝑖𝑔 𝑇𝑎𝑊

𝑎𝜇 − 𝑖𝑔 ′ 𝑌 𝐵𝜇, (1.1)

for a general field that transforms non-trivially under SU(3)c and SU(2)L,and has hypercharge 𝑌 . 𝑇𝑎 are the generators of weak isospin SU(2)L, with𝑇𝑎 = 1

2 𝜎𝑎 for the SM SU(2)L doublets, where 𝜎𝑎 (𝑎 = 1, 2, 3) are the Paulimatrices. 𝜆𝑎

2 (𝑎 = 1, 2, ..., 8) are the generators of SU(3)c, with 𝜆𝑎 the Gell-Mann matrices, which take care of the transformation of quarks as colortriplets.

The kinetic terms for the gauge bosons are:

ℒkin gauge = −14 𝐺

𝑎𝜇𝜈𝐺

𝜇𝜈𝑎 − 1

4 𝑊𝑎𝜇𝜈𝑊

𝜇𝜈𝑎 − 1

4 𝐵𝜇𝜈𝐵𝜇𝜈 , (1.2)

with the field strength tensors given by:

𝑊 𝑎𝜇𝜈 ≡ 𝜕𝜇𝑊

𝑎𝜈 − 𝜕𝜈𝑊

𝑎𝜇 + 𝑔 𝜖𝑎𝑏𝑐𝑊 𝑏

𝜇𝑊𝑐𝜈 , (1.3)

𝐺𝑎𝜇𝜈 ≡ 𝜕𝜇𝐺

𝑎𝜈 − 𝜕𝜈𝐺

𝑎𝜇 + 𝑔𝑠 𝑓

𝑎𝑏𝑐𝐺𝑏𝜇𝐺

𝑐𝜈 , (1.4)

𝐵𝜇𝜈 ≡ 𝜕𝜇𝐵𝜈 − 𝜕𝜈𝐵𝜇, (1.5)

where 𝜖𝑎𝑏𝑐 (𝑓𝑎𝑏𝑐) are the structure constants of SU(2)L (SU(3)c), i.e., forinstance for SU(2)L, [𝑇 𝑎, 𝑇 𝑏] = 𝑖𝜖𝑎𝑏𝑐𝑇 𝑐.

1.2 The Higgs mechanism 13

The fermions come in three generations or families, with identical quantumnumbers, but different masses.2 Let us focus for the moment on just thefirst family, which makes up all the visible matter of the Universe. Table 1.1summarizes the particle content of one SM family.

The SU(2)L charge is called weak isospin 𝑇 . The SM doublets (left-handed, 𝜓L = 1

2(1 − 𝛾5)𝜓), ℓ and 𝑄, have isospin 1/2. The third componentof isospin is 𝑇3 = +1/2 for the up components of the doublets, 𝜈L and 𝑢L,(𝑇3 = −1/2 for the down ones, 𝑑L and 𝑒L). The right-handed components(𝜓R = 1

2(1 + 𝛾5)𝜓) of the SM fermions, 𝑒R, 𝑢R and 𝑑R, are singlets, i.e., theydon’t transform under SU(2)L. Note that in the original SM (built whenneutrinos where thought to be massless) right-handed partners (𝜈𝑅) for theleft-handed neutrinos are absent. This is fine for anomaly cancellation withineach family, as the hypercharge of the hypothetical 𝜈R is zero.3 However, onemay wonder why wouldn’t neutrinos have right-handed partners.

The kinetic terms for the fermions are:

ℒkin fermions =∑𝛼

𝑖 𝜓𝛼 𝛾𝜇𝐷𝜇 𝜓𝛼, (1.6)

where 𝜓𝛼 = (𝑄, 𝑑R, 𝑢R, ℓ, 𝑒R).

1.2 The Higgs mechanismGauge symmetry explicitly forbids a mass term for bosons and fermions.These can be generated via spontaneous symmetry breaking (SSB) of the EWsymmetry. This is known as the Higgs mechanism. In the simplest version, a

Higgs field 𝜑 =(𝜑+

𝜑0

)is introduced as a complex scalar doublet under SU(2)L.

The most general Lagrangian for the Higgs field is:

ℒHiggs = (𝐷𝜇𝜑)†𝐷𝜇𝜑+ 𝜇2 𝜑†𝜑− 𝜆 (𝜑†𝜑)2 , (1.7)

where 𝜇2 > 0 gives a Mexican-hat type potential.To find the minimum of the Lagrangian, which will be at zero kinetic

energy, one minimizes the scalar potential 𝑉 (|𝜑|) = −𝜇2|𝜑|2 + 𝜆 |𝜑|4. 𝜑 = 0 isnot a minimum of the scalar potential due to the negative sign of 𝜇2. 𝑉 (|𝜑|) isminimised for |𝜑| =

√𝜇2/2𝜆 ≡ 𝑣√

2 . The measured VEV of the Standard Model

2Really at this point we are in the flavour basis, so only after diagonalising (going tothe mass basis), can one really speak about the masses of the fields. In other words, thedown quark with definite mass is a combination of the down quarks of the three families.

3If 𝐵 − 𝐿 is to be a gauge symmetry, 𝜈𝑅 are needed, as then the 𝐵 − 𝐿 gauge symmetryis no longer non-anomalous.


is 𝑣 ≃ 246 GeV [47]. Now we can rewrite the Higgs field, as an excitationaround this minimum:

𝜑(𝑥) = 𝑣 +𝐻(𝑥)√2

𝑒𝑖𝜎𝑎

2 𝜃𝑎(𝑥)/𝑣(

01

), (1.8)

with 𝑎 = 1, 2, 3. The 𝐻 component of the field 𝜑 is the Higgs boson, the onlyphysical particle of the scalar doublet, with mass 𝑚2

H = 2𝜆𝑣2. It has beenrecently discovered at the LHC, with a mass around 125 GeV [13, 14].

The 𝜃’s are the would-be Goldstone bosons, which become the longitudinaldegrees of freedom the massive gauge fields, the 𝑍 and the 𝑊 .

If we substitute eq. 1.8 into the kinetic term of the Higgs, (𝐷𝜇𝜑)†𝐷𝜇𝜑, weobtain the masses of the gauge bosons. The 𝑊 boson, 𝑊± ≡ 1√

2 (𝑊1 ∓ 𝑖𝑊2),has a mass 𝑚2

W ≡ 𝑔2 𝑣2

4 . However, there is mixing between 𝐵𝜇 and 𝑊 3𝜇 , so to

obtain the physical spectrum we have to diagonalize their mass matrix:

𝑣2

4(𝑊 3𝜇 𝐵𝜇

)⎛⎝ 𝑔2 −𝑔𝑔 ′

− 𝑔𝑔 ′ 𝑔 ′2

⎞⎠⎛⎝𝑊 𝜇3

𝐵𝜇

⎞⎠ , (1.9)

with the following rotation (using the short-hand notation 𝑠𝜃 ≡ sin 𝜃 and soon): (

𝑊 3𝜇

𝐵𝜇

)=(𝑐𝜃𝑊

𝑠𝜃𝑊

−𝑠𝜃𝑊𝑐𝜃𝑊

)(𝑍𝜇𝐴𝜇

),

with 𝑡𝜃𝑊≡ 𝑔 ′/𝑔, where 𝜃W is the Weinberg angle.

This matrix has a zero eigenvalue, which corresponds to the photon mass,the other massive particle being the 𝑍, with mass 𝑚2

Z ≡(𝑔2 + 𝑔 ′2

)𝑣2

4 .The photon is the only massless gauge boson, associated with an unbroken

subgroup of SU(2)R × U(1)Y: QED, U(1)em. From the covariant derivativewe can relate the electric charge 𝑄 to the third component of isospin 𝑇3 andto the hypercharge Y:

𝐷𝜇 ⊃ −𝑖 𝑔√2(𝜎+𝑊𝜇 + 𝜎−𝑊 *

𝜇

)− 𝑖 (𝑔 𝑐𝜃𝑊

𝑇3 − 𝑔 ′𝑌 𝑠𝜃𝑊)𝑍𝜇

− 𝑖𝑔𝑔 ′√

𝑔2 + 𝑔 ′2(𝑇3 + 𝑌 )𝐴𝜇 ,

where we assumed it acts on a SM SU(2)L doublet, with 𝜎+ = 𝜎1 + 𝑖 𝜎2,𝜎− = (𝜎+)†.

As we know from QED that the photon couples to the electron withstrenght 𝑒 (for 𝑄 = −1), we have that 𝑄 = 𝑇3 + 𝑌 and 𝑒 = 𝑔𝑔 ′√

𝑔2+𝑔 ′2.

1.3 Flavour: masses and mixings 15

1.3 Flavour: masses and mixingsCharged (only for non-trivial representations of SU(2)L, like the doublets ofthe SM) and neutral currents arise from the fermion kinetic terms, eq. 1.6.

For a general doublet ΨTL =

(𝑈L 𝐷L

), the neutral currents (NC) are:

ℒNC = 𝑔 𝑇 3𝜓 𝜓L 𝛾𝜇 𝜓L 𝑊

𝜇3 + 𝑔 ′(𝑄𝜓 −𝑇 3

𝜓)𝜓L 𝛾𝜇 𝜓L 𝐵𝜇 + 𝑔 ′𝑄𝜓 𝜓R 𝛾𝜇 𝜓R 𝐵

𝜇 =

= 𝑒 𝜓 𝛾𝜇

[𝑇 3𝜓

𝑠𝜃𝑊𝑐𝜃𝑊

12(1 − 𝛾5) −𝑄𝜓𝑡𝜃𝑊

]𝜓 𝑍𝜇 + 𝑒𝑄𝜓 𝜓𝛾𝜇𝜓𝐴

𝜇 ,

where 𝜓L represents either 𝑈L or 𝐷L. The last term is just the QED La-grangian.

And the charged currents (CC) are of the form:

ℒCC = 𝑔ΨL 𝛾𝜇(𝑇 1𝑊 𝜇

1 + 𝑇 2𝑊 𝜇2

)ΨL = 𝑒√

2 𝑠𝜃𝑊

𝑈L𝛾𝜇𝐷L 𝑊𝜇 + H.c., (1.10)

where notice that 𝑈L, 𝐷L are not the physical (massive) up- or down-typefermions.

The Higgs field provides fermion mass terms. With the following definitions𝜑 ≡ 𝑖𝜎2 𝜑

*, ℓ ≡ 𝑖𝜎2 ℓc, ℓ c ≡ 𝐶 ℓT 4 we can add to the SM Lagrangian (in

fact we have to add, as they are allowed by the symmetries) the followinginteractions:

ℒYukawa = ℓ𝑌𝑒𝑒R 𝜑 + ℓ𝑌𝜈𝜈R 𝜑 + 𝑄 𝑌𝑢𝑢R 𝜑 + 𝑄 𝑌𝑑 𝑑R 𝜑 + H.c. (1.11)

The Yukawa matrices 𝑌𝑒, 𝑌𝜈 , 𝑌𝑢, 𝑌𝑑 are 3 × 3 complex matrices. After SSB,for each fermion 𝑓 , with 𝑀𝑓 = 𝑌𝑓𝑣/

√2, we get:

ℒfermion mass =∑𝑓

(1 + 𝐻

𝑣

)𝑓L𝑀𝑓𝑓R + H.c. , (1.12)

with 𝑓 = 𝑢, 𝑑, 𝑒 (and 𝜈 if 𝜈R are present), and each of the fields understood asa vector in family or flavour space, i.e., 𝑢 = (𝑢, 𝑐, 𝑡), 𝑑 = (𝑑, 𝑠, 𝑏), 𝑒 = (𝑒, 𝜇, 𝜏)(and = (𝜈𝑒, 𝜈𝜇, 𝜈𝜏 ) if 𝜈R are present). The VEV sets the common scalefor the masses of the gauge bosons and the fermions, and the gauge andindividual Yukawa couplings provide the different hierarchies in mass betweenthe different particles (some of them like the top or the electron Yukawasspan ∼ 5 orders of magnitude!).

4Let us notice that the fundamental representation of SU(2) is pseudoreal.


We need to diagonalize these mass matrices. For the quark sector, forinstance for the up-type quarks we can perform the rotations 𝑢𝑚L = 𝑉𝑢L𝑢Land 𝑢𝑚R = 𝑉𝑢R𝑢R, where the superscript 𝑚 refers to mass eigenstates, suchthat we diagonalise the mass matrix: 𝑀𝑢 = 𝑉𝑢L𝐷𝑢𝑉

†𝑢R

. And similarly forthe down-type quarks: 𝑀𝑑 = 𝑉𝑑L𝐷𝑑𝑉

†𝑑R

, with 𝑑𝑚L = 𝑉𝑑L𝑑L and 𝑑𝑚R = 𝑉𝑑R𝑑R.After this diagonalization, an extra matrix, the Cabibbo-Kobayashi-Maskawa(CKM) quark mixing matrix, 𝑉CKM, will appear in the CC interactions:

ℒquark sectorCC = 𝑒√

2 𝑠𝜃𝑊

𝑢L𝛾𝜇 𝑉CKM𝑑L 𝑊𝜇 + H.c. (1.13)

where 𝑉CKM = 𝑉 †𝑢L𝑉𝑑L .5 Therefore, in the SM, while CC change flavour at

tree level, there are no tree level flavour changing neutral currents (FCNC).For massless neutrinos, the equivalent lepton mixing matrix disappears.

For instance, we can diagonalize the electron mass matrix trivially: 𝑀𝑒 =𝑉𝐿𝐷𝑒𝑉

†𝑅 , 𝑒R = 𝑉𝑅 𝑒R and ℓ = 𝑉𝐿ℓ.

However, for massive neutrinos, equivalently to the situation just describedfor the quark sector, CC that change flavour appear:

ℒlepton sectorCC = 𝑒√

2 𝑠𝜃𝑊

𝑒L𝛾𝜇𝑈PMNS𝜈L 𝑊𝜇 + H.c., (1.14)

with 𝑈PMNS the Pontecorvo-Maki-Nagakawa-Sakata (PMNS) lepton mixingmatrix.

For 3 generations, a 3 × 3 unitary matrix has 3 angles and 6 phases.However, 5 of the phases can be removed, as field redefinitions like 𝑓 → 𝑒𝑖𝛼𝑓𝑓leave the Lagrangian invariant. One cannot be removed (transforming allfields with the same phase, 𝛼), as it is associated to the conservation of baryonnumber. This way only 1 phase remains, is physical, and appears in the CKMmatrix. Regarding the lepton sector, the situation is analogous if neutrinosare Dirac particles, with only one phase present. If, however, neutrinos areMajorana particles, 2 extra physical phases appear in the PMNS mixingmatrix (see section 2.2, chapter 2).

As a summary, the SM final Lagrangian before SSB, which has all termspermitted by the gauge symmetry, particle content and renormalizability,6 isgiven by:

ℒSM = ℒkin fermions + ℒkin bosons + ℒYukawa + ℒHiggs, (1.15)5One can always start, for instance, in the up-quarks mass basis, as before SSB one

can perform independent unitary transformations in the ℓL and 𝑢L, and in this case𝑉CKM ≡ 𝑉𝑑L .

6As said in the introduction in part I, a term like 𝜃 𝜖𝜇𝜈𝜌𝜎 𝐺𝜇𝜈𝐺𝜌𝜎 is allowed and violatesCP.

1.4 Effective Field Theory 17

where different terms are given in eqs. 1.2, 1.6, 1.7 and 1.11. As explainedbefore, after SSB, charged and neutral currents, as well as mass terms forfermions and gauge bosons, are present.

1.4 Effective Field TheoryEffective field theory (EFT) is based on the well-tested principle that dynamicsat low energies (large distances) does not depend on the details of the dynamicsat high energies (short distances), up to a given accuracy.

The SM operators have dimension four or less, and are renormalizable.With EFT one can parametrize any high-energy physics at a high scale Λ byusing non-renormalizable operators, whose validity is restricted to energiessmaller than Λ. All the effective operators must respect Lorentz invariance andgauge symmetries, but do not need to respect global accidental symmetriesof the SM, like baryon and lepton number. The predictivity of the EFT, ofcourse, is restricted to some precision: we have to stop at some dimension, asthe number of possible efective operators is infinite. However, this does notpose any problem, as experimental signatures are also restricted to a givenprecision.

The Lagrangian of the effective operators can be parametrized as:

ℒeff = ℒ +∑𝑛=5

∑𝑖

(𝐶𝑛𝑖

Λ𝑛−4 𝒪𝑛𝑖 + H.c.

), (1.16)

where 𝑛 indicates the dimension of the operator and 𝑖 labels the differentoperators of a given dimension. The operators of dimension 𝑛 have dimen-sionful couplings with dimensions of inverse mass to the power 𝑛− 4, withthe coefficients 𝐶𝑛

𝑖 being dimensionless.7EFT will prove to be a very useful tool to analise Majorana neutrino

masses, i.e., lepton number violation (LNV) via effective operators. Forinstance, the Weinberg operator, which is the only dimension 5 operator withthe SM fields and violates lepton number (LN), and will be explained in detailin chapter 2, can be generated by different complete theories, such as theseesaw models. In this way, we can be very general in parameterizing differentMajorana neutrino mass models by making use of the Weinberg operator.

If one knows the full theory, the equivalence of both the effective theoryand the complete description, at the energy scale of the new particles (saythe mass of right-handed neutrinos in seesaw type I), is known as matching.

At dimension 6 there are many four fermion operators, and some of themlead to flavor changing neutral currents (FCNC), like the process 𝑏 → 𝑠𝛾, to

7The separation between Λ and 𝐶 is arbitrary, of course.


CP-violating electric dipole moments or to baryon number violating processes,like proton decay. To date none of these processes have been observed, settingthe scale of new physics Λ to be very high, depending on the particularoperator. If one assumes order one coefficients, from neutrino masses typicallyone gets Λ𝜈masses ∼ 𝒪(1013 GeV), from FCNC ΛFCNC ≥ 𝒪(10 − 103 TeV),while from proton decay Λp decay ≥ 𝒪(1016 GeV). These predictions are validfor a large class of NP models, and demonstrate the power of EFT. Of course,it could happen that the coefficients are not order one but suppressed (forinstance if they are generated at loops, or for some reason of the order of theYukawa of the electron), and then, for instance if one thinks of the Weinbergoperator as a parameterization of radiative and seesaw models, the LNV scalein these cases can be much lower.

This poses a problem of naturalness with the relevant operators of theSM:

∙ the Higgs mass, 𝑚H ≈ 125 GeV.

∙ the cosmological constant, observed to correspond to a vacuum energydensity 𝜌Λ ∼ 10−11eV4).8

Another place where naturality may play a role is the hypothetical right-handed mass 𝑚𝑅, although this case is slightly different as it is protected byLN, i.e., in its absence LN is a good symmetry, so it could be small in thet’Hooft sense (quantum corrections are proportional to itself, so if small attree level, it can be kept small naturally). However, if one thinks that it isnot protected by any gauge symmetry, it could be as large as the largest scalepresent in the theory, for instance the GUT scale, motivated by 𝑆𝑂(10).

Let us notice that all mass terms are in fact protected by another symmetry,scale invariance, that is anomalous, i.e., broken at the quantum level by therunning of the couplings.

8Any QFT estimation yields a value 60 − 120 orders of magnitude larger!, the worstdiscrepancy between theory and experiment in physics [48].

Chapter 2

Neutrino masses

We briefly review in this chapter the key facts about neutrinos, focusing ontheir masses. We refer the interested reader to [47, 49–51] for some moreextended reviews on neutrino physics and to [52] for a mandatory referenceon neutrino physics.

2.1 Brief history of neutrino oscillationsIn 1930, neutrinos were proposed by Pauli as neutral fermions with spin 1/2as an explanation to the apparent energy and spin non-conservation in nuclear𝛽 decay, which had a continuum spectrum typical of a two-body decay1 [53].

25 years later, Reines and Cowan discovered neutrinos escaping from anuclear reactor in Savannah River [54–56].2

In the late sixties, the Homestake experiment [57], directed by Davis,observed a discrepancy between the expected solar neutrino flux and the onedetected that led to the hypothesis that neutrinos are massive particles thatmix as a possible explanation [58].

In 1998, Super-Kamiokande reported evidence of neutrino oscillations inthe atmospheric neutrino flux and later SNO and KamLAND confirmed solarneutrino oscillations.

So the last two decades can be thought of as the neutrino golden era, withconfirmation that neutrinos are massive particles that mix thanks to the resultsobtained with solar [57, 59–62] and atmospheric [63–65] neutrinos, confirmedwith man-made sources: nuclear reactors [66] and accelerators [67,68].

However, currently the origin and nature of the neutrinos tiny mass is stilla mystery. Just one thing is known: the original SM with massless neutrinos

1We detect the electrons emitted, the electron anti-neutrinos are not detected.2Electron anti-neutrinos to be more precise.

20 Chapter 2. Neutrino masses

needs to be extended.

2.2 Neutrino parametersNeutrino oscillations require mixing among the three flavour states 𝜈𝛼, with𝛼 = 𝑒, 𝜇, 𝜏 , which can be expressed as quantum superpositions of threemassive states 𝜈𝑖, with 𝑖 = 1, 2, 3. The PMNS mixing matrix that appearsin charged current interactions, eq. 1.14, relates both basis (in the following𝑈PMNS ≡ 𝑈 for simplicity), i.e., 𝜈𝛼 = ∑

𝑖 𝑈𝛼𝑖𝜈𝑖. It can be parametrized interms of three Euler angles 𝜃12, 𝜃13, 𝜃23 and just a CP violating phase 𝛿 ifneutrinos are Dirac. However, there are two extra physical Majorana phases𝛼1, 𝛼2 if neutrinos are Majorana particles. The explicit realization of thelepton mixing matrix is: 𝑈 = ℛ · 𝒟, with 𝒟 = diag(𝑒𝑖 𝛼1/2, 𝑒𝑖 𝛼2/2, 1) and,using the short-hand notation cos 𝜃𝑖𝑗 ≡ 𝑐𝑖𝑗 and so on, ℛ is given by:

ℛ =

⎛⎜⎝ 𝑐12𝑐13 𝑠12𝑐13 𝑠13𝑒−𝑖𝛿

−𝑠12𝑐23 − 𝑐12𝑠23𝑠13𝑒𝑖𝛿 𝑐12𝑐23 − 𝑠12𝑠23𝑠13𝑒

𝑖𝛿 𝑠23𝑐13𝑠12𝑠23 − 𝑐12𝑐23𝑠13𝑒

𝑖𝛿 −𝑐12𝑠23 − 𝑠12𝑐23𝑠13𝑒𝑖𝛿 𝑐23𝑐13

⎞⎟⎠ .Oscillation experiments are only sensitive to mass-squared splittings

(Δ𝑚2𝑖𝑗 ≡ 𝑚2

𝑖 −𝑚2𝑗). Currently, there are two possible choices of the sign of

Δ𝑚231 ≡ Δ𝑚2

𝑎𝑡𝑚, while Δ𝑚221 ≡ Δ𝑚2

𝑠𝑜𝑙 is known to be positive. Therefore theordering of the masses is compatible with:

∙ 𝑚1 . 𝑚2 < 𝑚3 - Normal Hierarchy (NH).

∙ 𝑚3 < 𝑚1 . 𝑚2 - Inverted Hierarchy (IH).

∙ 𝑚1 ≈ 𝑚2 ≈ 𝑚3 - quasi-degenerate spectrum (QD). Notice that QD isstill allowed depending on the cosmological assumptions used to derivethe bound on the sum of the neutrino masses.

The minimum value for the sum of the masses from oscillation experiments is∼ 0.06 eV (∼ 0.1 eV) for NH (IH), while for QD it is larger.

So to describe neutrinos we need ≥ 6 different parameters: at least 2masses (data are still compatible with one being massless), 3 mixing anglesand 1 phase (2 more phases in the case of Majorana neutrinos). Noticethat recently the neutrino mixing angle 𝜃13 has been measured [71–73]. Wepresent in figure 2.1 the current values for the oscillation parameters, whichare extracted from a global fit from [69] (see also [74,75] for other global fits).

Regarding the absolute neutrino mass scale, the so-called effective elec-tron neutrino mass, 𝑚2

𝜈𝑒≡ ∑

𝑖𝑚2𝑖 |𝑈𝑒𝑖|2, is constrained by Tritium 𝛽-decay.

2.2 Neutrino parameters 21

NuFIT 1.2 (2013)

Free Fluxes + RSBL Huber Fluxes, no RSBL

bfp ±1σ 3σ range bfp ±1σ 3σ range

sin2 θ12 0.306+0.012−0.012 0.271 → 0.346 0.313+0.013

−0.012 0.277 → 0.355

θ12/◦ 33.57+0.77

−0.75 31.37 → 36.01 34.02+0.79−0.76 31.78 → 36.55

sin2 θ23 0.446+0.008−0.008 ⊕ 0.593+0.027

−0.043 0.366 → 0.663 0.444+0.037−0.031 ⊕ 0.592+0.028

−0.042 0.361 → 0.665

θ23/◦ 41.9+0.5

−0.4 ⊕ 50.3+1.6−2.5 37.2 → 54.5 41.8+2.1

−1.8 ⊕ 50.3+1.6−2.5 36.9 → 54.6

sin2 θ13 0.0231+0.0019−0.0019 0.0173 → 0.0288 0.0244+0.0019

−0.0019 0.0187 → 0.0303

θ13/◦ 8.73+0.35

−0.36 7.56 → 9.77 9.00+0.35−0.36 7.85 → 10.02

δCP/◦ 266+55

−63 0 → 360 270+77−67 0 → 360

∆m221

10−5 eV27.45+0.19

−0.16 6.98 → 8.05 7.50+0.18−0.17 7.03 → 8.08

∆m231

10−3 eV2(N) +2.417+0.014

−0.014 +2.247 → +2.623 +2.429+0.055−0.054 +2.249 → +2.639

∆m232

10−3 eV2(I) −2.411+0.062

−0.062 −2.602 → −2.226 −2.422+0.063−0.061 −2.614 → −2.235

Figure 2.1: Neutrino parameters from the global fit [69]. As the authorsstate, for Free Fluxes + RSBL reactor fluxes have been left free and shortbaseline reactor data (RSBL) (. 100 m) have been included included; forHuber Fluxes, no RSBL the flux predictions from [70] have been used, andRSBL data was not included.

However, cosmological observations provide the tightest constraints on theabsolute scale of neutrino masses (in particular on the sum of the masses,∑𝑖𝑚𝑖), via their contribution to the energy density of the Universe and the

growth of structure. In general these bounds depend on the assumptions madeabout the expansion history as well as on the cosmological data included inthe analysis [76], but typically ∑𝑖𝑚𝑖 < 1 eV (and in most analysis stronger,see table 2.1).

Finally, if neutrinos are Majorana particles, complementary informationon neutrino masses can be obtained from 0𝜈2𝛽 decay. The contribution ofthe known light neutrinos to the 0𝜈2𝛽 decay amplitude is proportional to theeffective Majorana mass of 𝜈𝑒, 𝑚𝑒𝑒 ≡ |∑𝑖𝑚𝑖𝑈

2𝑒𝑖|, which depends not only on

the masses and mixing angles of the mixing matrix but also on the phases.

All the experimental constraints coming from cosmology, Tritium and0𝜈𝛽𝛽 are summarized in table 2.1.


Tritium 𝛽 decay

𝑚𝜈𝑒 < 2.2 eV 95% CL [77,78]𝑚𝜈𝑒 < 2 eV 95% CL [79]

𝑚𝜈𝑒 (future) ∼ 0.2 eV 95% CL [80]

0𝜈𝛽𝛽

𝑚𝑒𝑒 < 0.2 − 0.4 eV 90% CL [81]𝑚𝑒𝑒 < 0.14 − 0.38 eV 90% CL [82]𝑚𝑒𝑒 < 0.34 eV 90% CL [83]

𝑚𝑒𝑒 (future) ∼ 0.01 eV 90% CL [84]

Cosmology∑𝑖𝑚𝑖 < 0.23 eV 95% CL [85]

Table 2.1: Current status and future prospects of the experimental constraintson neutrino parameters coming from cosmology, Tritium and 0𝜈𝛽𝛽 decay.

2.2.1 Unsolved questions in the neutrino sectorEven if it is true that in the last two decades there have been major discoveriesin the field of neutrinos, we still need further input from experiments, whosenext generation will try, among other things, to determine the following:

∙ The nature Dirac or Majorana of neutrinos, i.e., whether lepton number(LN) is a conserved symmetry or not (see next section). Observationof 0𝜈𝛽𝛽 would imply that LN is violated by two units, and thereforeneutrinos are Majorana particles [86].

∙ The sign of Δ𝑚223, i.e., whether the spectrum is NH or IH.

∙ The existence of CP violation in the lepton sector and its value.

∙ Lepton flavour violation (LFV) in the charged lepton sector, via obser-vation of processes like 𝜇 → 𝑒𝛾 (the current limit is 5.7 · 10−13 at 90%CL [87]).3

3As lepton flavour is violated in neutrinos (neutrino oscillations), it is also expectedto be violated at some point in the charged lepton sector. In the SM 𝜈L and 𝑒L comein 𝑆𝑈(2)L doublets, and as neutrinos are massive, after diagonalization CC interactionsappear, see 1.14.

2.3 Dirac and Majorana masses 23

∙ The absolute mass scale of neutrinos.

∙ If sterile neutrinos exist: LSND experiment reported evidence for theappearance of 𝜈𝑒 in a 𝜈𝜇 beam [88], that can be interpreted as oscillationswith additional sterile neutrinos with a mass-squared difference Δ𝑚2 ∼1 eV2. MiniBooNE neutrino and antineutrino data may be explainedand be compatible with LSND with several sterile neutrinos, by allowingfor CP violating effects between neutrino and antineutrino oscillations[89,90].

The hope is that combining these (and other) experimental results themechanism by which neutrino masses are generated will eventually be uncov-ered.

2.3 Dirac and Majorana massesFermions without electric charge, like neutrinos (only LN, which is a globalsymmetry in the SM), can have two classes of mass terms, Dirac and Majorana:

∙ The Dirac fermion has four internal degrees of freedom, describingparticles and antiparticles with opposite charges, each of them with twodifferent helicities. It is therefore appropriate for particles that carryinternal charges (such as electric charge) which distinguish them fromtheir counterparts, the antiparticles. The mass term connects oppositechirality fields (takes a left-handed particle 𝜓L to a right-handed particle𝜓R) and it has the following form:

𝑚D 𝜓R 𝜓L + H.c. (2.1)

∙ A Majorana fermion has two degrees of freedom, without any internal(conserved) charge, i.e., it is its own antiparticle, 𝜓𝑐 = 𝜓. The massterm takes a left-handed particle 𝜓L to a right-handed antiparticle 𝜓𝑐L,which is the CP conjugate of a left-handed particle, or takes a right-handed particle 𝜓R to a left-handed antiparticle, 𝜓𝑐R. It violates any𝑈(1) symmetry, like LN for the case of neutrinos, and it is of the form:

𝑚L 𝜓cL 𝜓L + H.c. (2.2)

A theory can have only left-handed neutrinos and their CP conjugatestates, without having right-handed neutrinos. In this case the neutrino couldbe massless or can have a Majorana mass 𝑚L like in eq. 2.2, which violates


LN by two units. However, active neutrinos 𝜈L come in SU(2)L doublets, ℓ, sowith the SM content 𝑚L 𝜈c

L 𝜈L is not gauge invariant, and therefore 𝑚L mustcome from a non-renormalizable operator, parametrized via the Weinbergoperator, see section 2.4. If the SM particle content is extended, 𝑚L can begenerated, as will be shown in chapter 3, either at tree level (seesaw models)or at loop level (radiative models).

When both the left- and right-handed neutrinos, 𝜈L and 𝜈R, are present,the most general mass terms are:

ℒ𝑀 = −12𝑚L𝜈𝑐L𝜈L −

12𝑚R𝜈𝑐R𝜈R −𝑚D𝜈𝑅𝜈L + H.c.,

and there are several possibilities:

∙ if all mass terms vanish neutrinos are massless, so they are two Weylspinors (ruled-out).

∙ If 𝑚L = 𝑚R = 0 the left-handed and the right-handed neutrinos combineto form a Dirac neutrino.4

∙ If 𝑚L = 0, 𝑚R = 0 and 𝑚D = 0, there are two Majorana neutrinos 𝜈Land 𝜈R with masses 𝑚L and 𝑚R.

∙ If 𝑚L or 𝑚R or both are non-vanishing, and also 𝑚D, the physical states(which are Majorana) are admixtures of the states 𝜈L and 𝜈R withmasses obtained by diagonalising the mass matrix, as will be shown inchapter 3, section 3.1.

So, if we add 𝜈R to the SM, which seems well-motivated by the fact thatall other fermions have left- and right-handed components, what is the mostnatural option, Dirac or Majorana masses? In QFT, we know that we have toadd all gauge invariant renormalizable terms compatible with the field contentof the theory, like 𝑚R 𝜈c

R 𝜈R, so we find that LN, which was an accidentalU(1) global symmetry of the SM, is violated.5 Moreover, if 𝜈R are added tothe SM, as they are gauge singlets (sterile), 𝑚R could be at an energy scalenot related to the EWS: for instance much larger than the EWS, leading tothe Seesaw mechanism that will be explained in sec. 3.1, but in principlethere is no reason for 𝑚R not to be, for example, at or below the EWS.

4Notice that a Dirac fermion can be regarded as two mass-degenerate Majorana fermionsof opposite CP.

5Also baryon number and generational lepton numbers are accidental global symmetriesof the SM. Any new neutrino mass model should violate generational lepton numbers,independently of their Dirac (LN conserving) or Majorana (lepton number violating, LNV)nature.

2.4 The Weinberg operator 25

On the other hand, to have Dirac neutrinos like all other fermions, wehave to forbid 𝑚R, that is, LN (or B-L6) has to be imposed by hand. LNwould be an exact symmetry, but not obtained accidentally from the gaugeinvariance and particle content as in the SM, but imposed by hand, whichis theoretically not very rewarding. In this case, neutrinos would be Diracparticles (𝜈 ≡ 𝜈L + 𝜈R), like the rest of the fermions.

In addition, for Dirac neutrinos, one has to face the quantitative problemof the values of their masses. As neutrinos would acquire their masses afterSSB like all other fermions, via the Higgs mechanism, their masses should beof the order of the vacuum expectation value (VEV) of the neutral componentof the Higgs field, 𝒪 (100 GeV), but they are at least 12 orders of magnitudesmaller! This problem of fine-tuning of Yukawas to give several orders ofmagnitude of difference in masses is also present for the other fermions,although quantitatively smaller.7 In the case of neutrinos, it is difficult toimagine that 173 and 10−11 GeV particles (the top and the neutrinos) acquiretheir masses via the same mechanism. In addition, all other fermions rangefrom the electron mass 𝒪 (1 MeV) up to the top mass, 𝒪 (100 GeV), whilethere would be a gap between neutrino masses 𝒪 (1 eV) and the electronmass.8

As will be explained in chapter 3, there can be explanations for thesmallness of their mass, like seesaw models (a heavy particle explains neutrinostiny mass) and radiative models (they are much lighter than other particlesbecause they are massless at tree level, with their mass generated at loops).In the next section we will study how if neutrinos are Majorana particles,their masses can be generally parametrized by an effective operator.

2.4 The Weinberg operatorAs we have argued in the previous section, it does not seem very naturalthat neutrinos are Dirac particles. LN is an accidental global symmetry ofthe SM (like baryon number) and if one thinks of the SM as a low-energyeffective theory (see section 1.4 for a brief discussion of EFT), one finds thatat dimension 5 there is only one effective operator, the Weinberg operator,

6Beyond perturbation theory LN is violated, however, B-L is preserved. Therefore, toavoid Majorana mass terms it is more appealing to impose by hand B-L, which is anomalyfree.

7As already said in the introduction I, for example the electron and the top masses are∼ 5 orders of magnitude away and altogether this is known as the flavour problem: whythree generations, why are the masses and mixings what they are...

8One can speculate whether it is the perfect range for 𝒪 (1 keV) right-handed neutrinos,which are in addition good warm dark matter candidates.


which happens to violate LN by two units:

ℒ5 =12𝑐𝛼𝛽ΛW

(ℓ𝛼𝜑) (𝜑†ℓ𝛽) + H.c. , (2.3)

where Λ𝑊 ≫ 𝑣 is the scale of new physics and 𝑐𝛼𝛽 are model-dependentcoefficients. Upon electroweak symmetry breaking, the Weinberg operatorleads to a Majorana mass matrix for the light neutrinos of the form

𝑚L = 𝑐𝑣2

Λ𝑊

. (2.4)

Notice that 𝑐𝛼𝛽 have flavour structure, and in some models can carryadditional suppression due to loop factors (as is the case in radiative mecha-nisms) and/or ratios of mass parameters (for instance in type II seesaw thereis a suppression of the LNV parameter 𝜇, see eq. 3.6, therefore 𝑐 ∝ 𝜇/𝑚𝜒).In those cases one typically relates directly ΛW, which does not need to beextremely large, to the masses of the new particles and absorb all suppressionfactors in 𝑐𝛼𝛽.

By doing some SU(2) Fierz transformations one can rewrite the Weinbergoperator in eq. (2.3):

(ℓ𝛼𝜑

) (𝜑†ℓ𝛽

)= −

(ℓ𝛼 ��𝜑

) (𝜑† ��ℓ𝛽

)= 1

2(ℓ𝛼 �� ℓ𝛽

) (𝜑† �� 𝜑

),

where 𝛼 and 𝛽 are family indices and �� ≡ (𝜎1, 𝜎2, 𝜎3) is a vector in the adjointrepresentation of SU(2) whose components are the 2×2 Pauli matrices. Fromthe three explicit forms one can easily read the three different new particlesthat can generate the Weinberg operator at tree-level: a hyperchargelessheavy fermion singlet (triplet) for the first (second) expressions, and a 𝑌 = 1heavy scalar triplet for the third one. These are precisely seesaw models:type I (III) and II respectively, which we discuss in more detail in chapter 3.This demonstrates the power of the Weinberg operator, which parametrizesMajorana neutrino masses of very different extensions of the SM.

Chapter 3

Neutrino mass models

There are many neutrino mass models on the market. In this chapter webriefly review some of the most popular ones. We start by discussing typeI seesaw, which requires the addition of right-handed neutrinos to the SM,which was introduced in section 2.3 of chapter 2. We also summarize othertypes of seesaw models, before moving on to some of the best-motivatedhigh-energy frameworks. We conclude with radiative models, which will bethe main topic of this part of the thesis.

3.1 Type I: fermionic singletsIn type I seesaw [91–95], 𝑛 SM fermionic singlets are added to the SM; thesehave the quantum numbers of right-handed neutrinos, and can be denoted by𝜈R𝑖. Note that to explain neutrino data, which requires al least two massiveneutrinos, a minimum of two extra singlets are needed; however, 𝑛 = 3 isaesthetically (and based on a possible 𝑆𝑂(10) unification) the most reasonableoption. Having no charges under the SM, Majorana masses for right-handedneutrinos are allowed by gauge invariance, so the new terms in the Lagrangianare:

ℒ𝜈R = 𝑖 𝜈R𝛾𝜇𝜕𝜇𝜈R −

(12𝜈

cR𝑚R𝜈R + ℓ 𝜑 𝑌 𝜈R + H.c.

), (3.1)

where 𝑚R is a 𝑛 × 𝑛 symmetric matrix, 𝑌 is a general 3 × 𝑛 matrix andwe have omitted flavour indices for simplicity. After spontaneous symmetrybreaking (SSB), the neutrino mass terms are given by (see the diagram offigure 3.1)

ℒ𝜈 mass = −12(𝜈L 𝜈c

R

) ( 0 𝑚D𝑚T

D 𝑚R

) (𝜈c

L𝜈R

)+ H.c. , (3.2)

where 𝑚D = 𝑌 𝑣√2 .

28 Chapter 3. Neutrino mass models

ℓL

φ

ℓL

φ

νR,ΣcL

Y YM

Figure 3.1: Seesaw type I neutrino mass diagram.

𝑚R is in principle free, however if 𝑚R ≫ 𝑚D, upon block-diagonalizationone obtains 𝑛 heavy leptons which are mainly SM singlets, with masses ∼ 𝑚R,and the well-known seesaw formula for the effective light neutrino Majoranamass matrix,

𝑚𝜈 ≃ −𝑚D 𝑚−1R 𝑚T

D , (3.3)which naturally explains the smallness of light neutrino masses as a con-sequence of the presence of heavy SM singlet leptons. Notice that if onewants leptogenesis to be the source of the baryon asymmetry of the Universe,typically there is a lower bound on these right-handed neutrinos, 𝑚R > 109

GeV [22].

3.2 Type II: scalar tripletThe type II seesaw [96–100] only adds to the SM field content one scalar tripletwith hypercharge 𝑌 = 1 and assigns to it lepton number 𝐿 = −2. In thedoublet representation of SU(2)L, using �� = (𝜎1, 𝜎2, 𝜎3) and �� = (𝜒1, 𝜒2, 𝜒3),the triplet can be written as a 2 × 2 matrix, whose components are

𝜒 = 1√2�� · �� =

(𝜒+ 𝜒++

𝜒0 −𝜒+

), (3.4)

where we defined 𝜒++ ≡ 𝜒1−𝑖𝜒2√2 , 𝜒+ ≡ 𝜒3√

2 and 𝜒0 ≡ 𝜒1+𝑖𝜒2√2 , with third

component of isospin 𝑇3 = 1, 0,−1, as can be seen by computing [𝜎3/2, 𝜒].Gauge invariance allows a Yukawa coupling of the scalar triplet to two

lepton doublets,

ℒ𝜒 =((𝑌 †

𝜒 )𝛼𝛽 ℓ𝛼𝜒ℓ𝛽 + H.c.)

− 𝑉 (𝜑, 𝜒) , (3.5)

where 𝑌𝜒 is a symmetric matrix in flavour space. The scalar potential has,among others, the following terms:

𝑉 (𝜑, 𝜒) = 𝑚2𝜒 Tr[𝜒𝜒†] −

(𝜇𝜑†𝜒†𝜑+ H.c.

)+ . . . (3.6)

3.3 Type III: fermionic triplets 29

ℓL ℓL

χ

φ φ

Y

µ

Figure 3.2: Seesaw type II neutrino mass diagram.

The 𝜇 coupling violates lepton number explicitly, and it induces a vacuumexpectation value (VEV) for the triplet via the VEV of the doublet, even if𝑚𝜒 > 0. In the limit 𝑚𝜒 ≫ 𝑣 this VEV can be approximated by:

⟨𝜒⟩ ≡ 𝑣𝜒 ≃𝜇𝑣2

2𝑚2𝜒

; (3.7)

then, the Yukawa couplings in equation (3.5) lead to a Majorana mass matrixfor the left-handed neutrinos via the diagram of figure 3.2:

𝑚𝜈 = 2𝑌𝜒𝑣𝜒 = 𝑌𝜒𝜇𝑣2

𝑚2𝜒

. (3.8)

The dependence of neutrino masses on 𝑌𝜒 and 𝜇 can be understood fromthe Lagrangian, since the breaking of LN results from the simultaneouspresence of the Yukawa and 𝜇 couplings. As long as 𝑚2

𝜒 is positive and large,𝑣𝜒 will be small, in agreement with the constraints from the 𝜌 parameter,𝑣𝜒 . 6 GeV [101]. Moreover, the parameter 𝜇, which has dimensions of mass,can be naturally small, because in its absence LN is recovered, increasing thesymmetry of the model.

3.3 Type III: fermionic tripletsIn the type III seesaw model [102,103], the SM is extended by fermion 𝑆𝑈(2)Ltriplets Σ𝛼 with zero hypercharge. As in type I, at least two fermion tripletsare needed to have two non-vanishing light neutrino masses. We choosethe spinors Σ𝛼 to be right-handed under Lorentz transformations and write


them in SU(2) Cartesian components Σ𝛼 = (Σ1𝛼,Σ2

𝛼,Σ3𝛼). The Cartesian

components can be written in terms of charge eigenstates as usual

Σ+𝛼 = 1√

2(Σ1

𝛼 − 𝑖Σ2𝛼) , Σ0

𝛼 = Σ3𝛼 , Σ−

𝛼 = 1√2

(Σ1𝛼 + 𝑖Σ2

𝛼) , (3.9)

and the charged components can be further combined into negatively chargedDirac fermions 𝐸𝛼 = Σ−

𝛼 + Σ+c𝛼 . Using standard four-component notation the

new terms in the Lagrangian are given by

ℒΣ = 𝑖 Σ𝛼𝛾𝜇𝐷𝜇 · Σ𝛼 −

(12 (𝑚R)𝛼𝛽Σc

𝛼 · Σ𝛽 + 𝑌𝛼𝛽 ℓ𝛼(�� · Σ𝛽

)𝜑+ H.c.

),

(3.10)where 𝑌 is the Yukawa coupling of the fermion triplets to the SM leptondoublets and the Higgs, and 𝑚R their Majorana mass matrix, which can bechosen to be diagonal and real in flavour space.

After SSB the neutrino mass matrix can be written as

ℒ𝜈 mass = −12(𝜈L Σ0c

) ( 0 𝑚D𝑚T

D 𝑚R

) (𝜈c

LΣ0

)+ H.c. , (3.11)

which is the same as in the type I seesaw just replacing 𝜈R𝑖 by Σ0𝛼, and

therefore leads to a light neutrino Majorana mass matrix

𝑚𝜈 ≃ −𝑚D 𝑚−1R 𝑚T

D . (3.12)

However, since the triplet has also charged components with the same Ma-jorana mass, in this case there are stringent lower bounds on the new massscale from direct searches, so 𝑚R & 100 GeV.

3.4 Inverse seesawIn inverse and linear seesaw models, one can have much lighter right-handedneutrinos masses than in type I seesaw (although in that case right-handedneutrinos can also be light if their Yukawas are small) and still get the correctneutrino mass scale [104]. In these models, LN is slightly broken by a smallparameter protected from radiative corrections, and flavour and CP violationeffects are not suppressed by light neutrino masses. The model incorporatestwo singlet fermions, 𝑁 and 𝑆, per generation, to which we assign LN one.In the (𝜈L, 𝑁

𝑐, 𝑆) basis the 9 × 9 mass matrix Ω of the neutral sector afterSSB has this form:

3.5 High-energy frameworks 31

Ω =

⎛⎜⎝ 0 𝑚D 𝑀𝑇L

𝑚𝑇D 0 𝑀𝑇

𝑀L 𝑀 𝜇

⎞⎟⎠,

where 𝑚D = 𝑌 𝑣/√

2, 𝑀 are 3×3 arbitrary complex matrices in flavour spaceand 𝜇 is complex and symmetric. If 𝑀L = 0 and 𝜇 = 0, the models is calledinverse seesaw, while if 𝑀L = 0 and 𝜇 = 0 it is called linear seesaw. LN isviolated in the 𝜇 and or 𝑀L terms.1

Let us focus on the inverse seesaw limit, 𝑀L = 0. If we assume 𝑚D, 𝜇 ≪𝑀 , the matrix Ω can be diagonalized by a unitary transformation leadingto 9 mass eigenstates: three light neutrinos, while the other three pairs oftwo-component leptons combine to form three quasi-Dirac leptons. We canblock-diagonalize first the 2 − 3 sub-matrix, and then the 1 − 2. The effectiveMajorana mass matrix for the light neutrinos is approximately given by:

𝑚𝜈 = 𝑚𝑇D(𝑀𝑇 )−1𝜇𝑀−1𝑚D, (3.13)

while the three pairs of heavy neutrinos have masses of order 𝑀 , and theadmixture among singlet and doublet 𝑆𝑈(2) states is suppressed by 𝑚D/𝑀 .Although 𝑀 is a large Dirac (Δ𝐿 = 0) mass scale suppressing the lightneutrino masses, it can be much smaller than in type I seesaw, as there is thefurther suppression of the LNV scale 𝜇/𝑀 . In the limit 𝜇 = 0, LN would beconserved, and the three light neutrinos would be massless Weyl particles,while the six heavy neutral leptons will combine to form three Dirac fermions.

3.5 High-energy frameworks

3.5.1 Left-right symmetric modelsSome possible high energy well-motivated theories that embed seesaw mech-anisms are left-right (LR) symmetric models, which restore parity at highenergies [95, 105,106], based on the group 𝑆𝑈(2)L × 𝑆𝑈(2)R × 𝑈(1)B−L.

In LR symmetric models, the right-handed fermion fields are doubletsunder 𝑆𝑈(2)R. There are both 𝑊L and 𝑊R, and left- and right-handedneutrinos, 𝜈L and 𝜈R, so neutrinos become massive.

The minimal model involves a bi-doublet 𝜑(2, 2, 0) and a pair of tripletsΔL(3, 1, 2) and ΔR(1, 3, 2). When the LR and the EW symmetries are broken,the bidoublets provide fermion masses, while the triplets provide Majoranamass terms for 𝜈L and 𝜈𝑅.

1If LN was spontaneously broken by the VEV of a singlet ⟨𝜎⟩, 𝜇 = 𝑦 ⟨𝜎⟩.


The minimum of the potential is achieved for 𝑣2 ∼ 𝑣L𝑣R [107], with 𝑣L(𝑣R) the VEV of the triplet scalar ΔL (ΔR). Thus, if 𝑣R ≫ 𝑣L (in agreementwith the 𝜌 parameter, see discussion on 𝑣𝜒 ≡ 𝑣𝐿 in section 3.2) the Majoranamass term for 𝜈R (𝑚R = 𝑓 𝑣R) is large and the Majorana mass term for theleft-handed neutrinos (𝑚L = 𝑓 𝑣L) is small, where the couplings 𝑓 are thesame as there is a left-right symmetry. Therefore, the neutrino masses aregiven by:

𝑚𝜈 = 𝑓 𝑣L −𝑚D (𝑓 𝑣R)−1 𝑚𝑇D, (3.14)

and can be naturally small.

3.5.2 Supersymmetric models

There is an intrinsically supersymmetric way of breaking LN by breakingthe so-called R parity, which is defined as R = (−1)3B+L+2S, with B (L) thebaryon (lepton) number and S the spin, so SM (supersymmetric) particleshave 𝑅 = 1(−1) [108–117] (for a review see [118]). In this scenario, theSM doublet neutrinos mix with the neutralinos, i.e., the supersymmetric(fermionic) partners of the neutral gauge and Higgs bosons. As a consequence,Majorana masses for neutrinos (generated at tree level and at one loop) arenaturally small because they are proportional to the small R-parity-breakingparameters.

3.5.3 Grand Unified Theories

Some of these left-right models and supersymmetric extensions of the SMare naturally embedded in Grand Unified Theories (GUTs) like SO(10) orSU(5) [33], although this is not needed. It is remarkable the fact that thehypothetical right-handed neutrinos, 𝜈R, would complete the representation16 of SO(10), so that all fermions of each family are contained in a singlerepresentation of the unifying group. This looks too impressive not to besignificant.

We refer the reader to [52] for a more thorough discussion on how toaccommodate neutrino masses in high energy frameworks.

3.6 Radiative models 33

Figure 3.3: The Zee model diagram contributing to neutrino masses.

3.6 Radiative modelsSmall Majorana neutrino masses may also be induced by radiative corrections.Typically, on top of loop factors of at least 1/(4𝜋)2,2 there are additionalsuppressions due to couplings and/or ratios of masses, leading to the observedlight neutrino masses with a NP scale not far above the EWS, so in principlethese models can be tested at the LHC.

In this thesis we will analyse in detail various radiative models, like theZee-Babu model [119,120], and another two-loop mechanism [121], by addingnew generations (see the scientific research, part V). Therefore, here we justgive a brief introduction to two other examples: the Zee model [119, 122] andthe Inert Doublet Model [123–125].

3.6.1 The Zee ModelThe Zee model [119, 122] uses the fact that ℓℓ couples to a charged scalarsinglet 𝜒+ with 𝑌 = 1. This coupling, call it 𝑓 , is anti-symmetric. Two Higgsdoublets are needed so the LNV term 𝜑2𝜑1 𝜒

− exists.If 𝜑2 does not couple to leptons, the diagram shown in figure 3.3 gives

neutrino masses at one loop, and the final expression of the neutrino massmatrix can be written as [126]:

(𝑚𝜈)𝑖𝑗 = 𝐶 𝑓𝑖𝑗(𝑚2ℓ𝑖 −𝑚2

ℓ𝑗), (3.15)

where 𝐶 is some constant that includes the loop factor and 𝑚ℓ are the chargedleptons masses. Therefore, the mass matrix is symmetric, as it has to be, andthe 𝑒𝑒 element is suppressed with respect to the other ones, which rendersin general a tiny rate of 0𝜈𝛽𝛽. The reader is referred to [127] for a recentanalysis of the model.

2More than three loops typically yield too light neutrino masses or have problems withLFV.


3.6.2 The Scotogenic modelThe Scotogenic model3 (see for instance [128] for the original proposal, and[123–125, 129–136] for some among the long list of different studies of themodel, some of them with just a inert doublet and no inert singlet fermions)is an extension of the SM which provides both neutrino masses and a darkmatter (DM) candidate.

The SM is extended with a second scalar doublet 𝜂 and at least tworight-handed singlet (or triplet) fermions 𝜈R𝑖

4 (or Σ𝑖) with masses 𝑚R𝑖 areneeded to give masses to the two known active massive neutrinos.5

All of the new fields are odd under a discrete symmetry 𝑍2, while theSM particles are even. The lightest of them is stable, so if it is the neutralcomponent and if its mass and interactions are the correct ones to have themeasured relic abundance, it could provide the DM of the Universe.

First of all, let us briefly explain how a discrete symmetry can be motivated,although in principle it looks ad hoc. A 𝑍N discrete symmetry can be gaugedin a U(1)D group. For example, consider matter fields 𝜒 with U(1)D charge𝑄𝜒 = −1 and Higgs field 𝜑 with charge 𝑄𝜑 = 𝑁. Let us call 𝑣 the VEV ofthis Higgs field which breaks spontaneously the symmetry. Then we can havethat the symmetry is broken by the VEV:

ℒ = 𝜑𝜒𝑁 + H.c. → 𝑣𝜒𝑁 + H.c., (3.16)

and we see how the Lagrangian has a discrete remnant 𝑍N symmetry:

𝜒 → 𝑒𝑖2𝜋/𝑁𝜒,

that it is exactly conserved due to its gauge origin.

The new interactions of the model that are added to the SM are:

ℒ𝜈R = 𝑖 𝜈R𝛾𝜇𝜕𝜇𝜈R −

(12𝜈

cR𝑚R𝜈R + ℓ 𝜂 𝑌 𝜈R + H.c.

), (3.17)

and the most general scalar potential 𝑉 (𝜂, 𝜑) is:

𝑉 (𝜂, 𝜑) = 𝑚21 𝜑

†𝜑+𝑚22 𝜂

†𝜂 +12𝜆1(𝜑†𝜑)2 +

12𝜆2(𝜂†𝜂)2+

3Also sometimes known as the Inert Doublet Model (IDM).4We call them 𝜈R𝑖, although they are not what one normally means by right-handed

neutrinos: they do not couple to the lepton doublets via the Higgs, i.e., there is no Diracmass after SSB.

5If the three of them are massive we would need to add three right-handed singlets, asin seesaw models.

3.6 Radiative models 35

Figure 3.4: Scotogenic model diagram contributing to neutrino masses. The𝑁 represent either right-handed neutrinos, 𝜈R𝑖, or the neutral component offermion triplets, Σ0.

+ 𝜆3(𝜑†𝜑)(𝜂†𝜂) + 𝜆4(𝜑†𝜂)(𝜂†𝜑) +12𝜆5

((𝜑†𝜂)2 + H.c.

). (3.18)

Notice that there is a U(1) global symmetry (LN) violated by the simulta-neous presence of 𝑚R, 𝜆5 and 𝑌 . If we assume 𝑚2

2 > 0 so ⟨𝜂⟩ = 0, the 𝑍2 isexactly conserved and neutrinos remain massless at tree level. The masses ofthe scalar particles in this model are:

𝑚2𝐻 = −2𝑚2

1 = 2𝜆1𝑣2, (3.19)

𝑚2𝜂+ = 𝑚2

2 + 𝜆3 𝑣2/2, (3.20)

𝑚2𝜂0

1= 𝑚2

2 + (𝜆3 + 𝜆4 + 𝜆5) 𝑣2/2, (3.21)

𝑚2𝜂0

2= 𝑚2

2 + (𝜆3 + 𝜆4 − 𝜆5) 𝑣2/2, (3.22)

where 𝐻 is the SM Higgs and the rest are the components of 𝜂. The lightestof 𝜂0

1, 𝜂02 can serve as a DM candidate (if all 𝜈R𝑖 are heavier). Notice that

the mass splitting between 𝜂01 and 𝜂0

2 is proportional to 𝜆5, which is naturallysmall, as in its absence LN is conserved.

The one-loop diagram that contributes to neutrino masses from 𝜂01 and 𝜂0

2exchange is shown in figure 3.4, and its exact contribution is given by:

(𝑚𝜈)𝑖𝑗 =∑𝑘

𝑌𝑖𝑘 𝑌𝑗𝑘𝑚R𝑘

16 𝜋2

⎡⎣ 𝑚2𝜂0

1

𝑚2𝜂0

1−𝑚2

R𝑘ln𝑚2𝜂0

1

𝑚2R𝑘

−𝑚2𝜂0

2

𝑚2𝜂0

2−𝑚2

R𝑘ln𝑚2𝜂0

2

𝑚2R𝑘

⎤⎦ . (3.23)

In the limit 𝑚2𝜂0

1−𝑚2

𝜂02

= 𝜆5𝑣2 ≪ 𝑚2

0 = (𝑚2𝜂0

1+𝑚2

𝜂02)/2, we get:

(𝑚𝜈)𝑖𝑗 = 𝜆5𝑣2

16 𝜋2

∑𝑘

𝑌𝑖𝑘 𝑌𝑗𝑘𝑚R𝑘

𝑚20 −𝑚2

R𝑘

[1 −

𝑚2R𝑘

𝑚20 −𝑚2

R𝑘ln

𝑚20

𝑚2R𝑘

]. (3.24)

There are some limiting cases:


∙ For 𝑚2R𝑘 ≪ 𝑚2

0:

(𝑚𝜈)𝑖𝑗 =𝜆5𝑣

2

8𝜋2𝑚20

∑𝑘

𝑌𝑖𝑘 𝑌𝑗𝑘𝑚R𝑘. (3.25)

∙ For 𝑚2R𝑘 ≫ 𝑚2

0:


2

8𝜋2

∑𝑘

𝑌𝑖𝑘 𝑌𝑗𝑘

𝑚R𝑘

[−1 − ln

𝑚20

𝑚2R𝑘

]. (3.26)

∙ For 𝑚2R𝑘 ≃ 𝑚2

0:


2

16 𝜋2

∑𝑘

𝑌𝑖𝑘 𝑌𝑗𝑘

𝑚R𝑘. (3.27)

So the typical seesaw scale is reduced by ∼ 𝜆5(4𝜋)2 . This means that for

𝜆5𝑌2 ≃ 10−8, we can have neutrino masses with a NP scale at the TeV.

If we consider 𝜈R𝑖 lighter than the 𝜂’s and we want it to be a DM candidateand at the same time to give the right order of magnitude of neutrino masses,there are strong constraints coming from LFV processes such as 𝜇 → 𝑒 𝛾.

However, as studied in [129, 130] it is still possible to find compatiblepoints that satisfy all constraints (for instance with 𝜆5 ∼ 10−9). Recently, itwas also studied in [136] the possibility that the lightest right-handed singletdoes not reach thermal equilibrium in the early Universe so that it behaves asa Feebly Interacting Massive Particle (FIMP). For the case of fermion triplets,as they have gauge interactions, it is in principle much easier to satisfy thethree constraints: neutrino masses, DM abundance and the LFV constraints.

For 𝜂 lighter than 𝜈R𝑖, as it is a doublet, it is possible to satisfy thestrong constraints coming from relic abundance, LFV, direct detection and soon [131,132]. In this case, there are also constraints on 𝜆5 coming from DMrelic abundance and direct detection bounds, see for instance [133]. Studiesof the scotogenic model in the context of leptogenesis and baryogenesis canbe found in [134] and [135], respectively.

We emphasize here that the Scotogenic model is a nice example of aradiative model that connects issues in principle disconnected such as neutrinosmasses and DM, with a powerful phenomenology.

Part III

Dark matter

Chapter 4

The Cosmological Model in anutshell

In this chapter we briefly review the Cosmological Model, before focusing ondark matter (DM) in the next chapter.

4.1 The basic ingredients of the ΛCDM modelThe Cosmological Model is also known as the ΛCDM model, where Λ refersto the cosmological constant, which causes the accelerated expansion of theUniverse we have observed, and CDM stands for cold dark matter (it couldalso be warm, see chapter 5), which is an extra matter component in additionto the luminous one, that we observe via its gravitational effects.

These are the basic ingredients of the ΛCDM model:

∙ The Cosmological Principle. By assuming that the symmetries of theUniverse, thought of as a perfect fluid, are isotropy (invariance underrotations) and homogeneity (invariance under translations), on scaleslarger that ∼ 100 Mpc, one obtains that the metric 𝑔𝜇𝜈 , known as theFriedmann-Lemaitre-Robertson-Walker (FLRW) metric, is:1

𝑑𝑠2 ≡ 𝑔𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 = −𝑑𝑡2 + 𝑎2(𝑡)

[𝑑𝑟2

1 − 𝑘𝑟2 + 𝑟2𝑑Ω], (4.2)

1Recall Minkowski’s metric for a flat space-time:

𝑑𝑠2 = −𝑑𝑡2 + 𝑑𝑟2 + 𝑟2𝑑Ω2. (4.1)

40 Chapter 4. The Cosmological Model in a nutshell

with 𝑑Ω = 𝑑𝜃2 + sin2 𝜃𝑑𝜑2, 𝑎(𝑡) the scale factor which parametrizes theevolution of space-time and 𝑘 the curvature of space-time (𝑘 = 0, seebelow).

∙ Einsteins’ Field Equations of General Relativity (GR), invariant undergeneral transformations of coordinates, which are:

𝑅𝜇𝜈 − 12𝑔𝜇𝜈𝑅 + Λ𝑔𝜇𝜈 = 8𝜋𝐺

𝑐4 𝑇𝜇𝜈 , (4.3)

where 𝑔𝜇𝜈 is the space-time metric, 𝑅𝜇𝜈 is the Ricci tensor, 𝑅 = 𝑔𝜇𝜈𝑅𝜇𝜈

the Ricci scalar and 𝑇𝜇𝜈 is the energy-momentum tensor, constructed interms of the matter Lagrangian density. As already mentioned, Λ is thecosmological constant, which parametrizes the accelerated expansion ofthe Universe. A pictorial representation of GR is that energy (matter)tells space-time how to curve, while space-time tells matter how to move.Eq. 4.3 can be derived from the Hilbert-Einstein action (with 𝑔 =det 𝑔𝜇𝜈):

𝑆 =∫ √

−𝑔𝑑4𝑥(

𝑐4

16𝜋𝐺(𝑅 − 2Λ) + ℒ𝑚

), (4.4)

through the Principle of Least Action (𝛿𝑆 = 0). ℒm is the mattercontent of the theory, whose variation with respect to the metric givesrise to the energy-momentum tensor 𝑇𝜇𝜈 of eq. 4.3.In principle ℒm can be associated to the content of the Standard Model(SM) of particle physics, plus any beyond the SM (BSM) physics, suchas dark matter (DM), neutrino masses, inflation and so on, i.e.:2

ℒm = ℒSM + ℒBSM = ℒSM + ℒ𝜈masses + ℒDM + ... (4.5)

However, in general, the different energy components are interpreted asforming a fluid, and are divided in matter and radiation. For a perfectfluid we have that:

𝑇 𝜇𝜈 = (𝜌+ 𝑝)𝑈𝜇𝑈𝜈 + 𝑝𝑔𝜇𝜈 , (4.6)

with 𝜌 and 𝑝 the energy density and the pressure and 𝑈𝜇 the four-velocity vector of the fluid. In an homogeneous and isotropic Universe,it is given by:

𝑇 𝜇𝜈 = diag(−𝜌, 𝑝, 𝑝, 𝑝). (4.7)2Let us notice that GR, encoded in eq. 4.3, is a classical theory, while the SM is a

Quantum Field Theory (QFT). A more complete theory which would unify both gravity andand gauge interactions is still missing, as quantization of GR yields a non-renormalizabletheory. String Theory, that goes beyond QFT, is the most popular candidate.

4.2 Friedmann equations 41

∙ The equation of state of different energy components, that relatestheir pressure and their density: 𝑝𝑖 = 𝑤𝑖 𝜌𝑖. From the First Law ofThermodynamics, 𝑑𝐸𝑖 + 𝑑𝑊𝑖 = 0, with the energy being 𝐸𝑖 = 𝜌𝑖𝑉 andthe work done by the system 𝑑𝑊𝑖 = 𝑝𝑖𝑑𝑉 , with 𝑉 = 𝑎3, we get:

��𝑖𝜌𝑖

= −3(1 + 𝑤𝑖)��

𝑎, (4.8)

which can be integrated to obtain:

𝜌𝑖 ∝ 𝑎−3(1+𝑤𝑖) . (4.9)

Matter has 𝑤matter = 0, so 𝜌matter ∝ 𝑎−3. For radiation 𝑤rad = 1/3 andtherefore 𝜌rad ∝ 𝑎−4, and for the cosmological constant 𝑤Λ = −1 andtrivially 𝜌Λ = constant.

4.2 Friedmann equationsUsing the metric 𝑔𝜇𝜈 of eq. 4.2 in Einsteins equations of GR, eq. 4.3,together with eq. 4.7, one obtains the Friedmann equations, which describethe evolution of the Universe in terms of the energy content:(

��

𝑎

)2= 8𝜋𝐺

3 𝜌tot(𝑡) − 𝑘

𝑎2 (4.10)

and (��

𝑎

)2= −4𝜋𝐺

3 (𝜌tot(𝑡) + 3𝑝tot), (4.11)

where 𝜌tot = ∑𝑖 𝜌𝑖 and 𝑝tot = ∑

𝑖 𝑝𝑖. Using eq. 4.9 (which can also be obtainedcombining eqs. 4.10 and 4.11) in eq. 4.10 (𝑘 = 0), one obtains the timeevolution of the scale factor:

𝑎(𝑡) ∝ 𝑡1/2, (4.12)

for the epoch of radiation domination (RD), i.e., when the energy density ofthe Universe was dominated by radiation, at 𝑇 > 𝑇M−R equality ∼ 1 eV, and

𝑎(𝑡) ∝ 𝑡2/3 (4.13)

for the matter-dominated epoch (MD), i.e., when the energy density of theUniverse was dominated by matter.

For a cosmological constant (that dominates today), 𝜔 = −1, one gets:

𝑎(𝑡) ∝ 𝑒𝐻0𝑡. (4.14)


Similarly, an accelerated expansion happens during inflation, see section4.5.

The left-hand side of eq. 4.10 is the square of the Hubble parameter,𝐻(𝑡) ≡ ��

𝑎. Its value today is 𝐻0 = (73.8 ± 2.4) km s−1 Mpc−1 as measured by

the Hubble Space Telescope [137]3; recently, however, the Planck satellite hasmeasured a smaller value: 𝐻0 = (67.3 ± 1.2) km s−1 Mpc−1 [85].

From observations of the Cosmic Microwave Background (CMB) - therelic photons that were released when electrons and protons combined to formhydrogen atoms - made by the Planck satellite [85], we know that the energydensity is to a great accuracy equal to the critical density, 𝜌crit: the energydensity needed to have a flat Universe (𝑘 = 0), defined from eq. 4.10 as:

𝜌crit ≡ 3𝐻20

8𝜋𝐺 ∼ 104 ℎ2 eVcm3 ∼ 3 · 1010 MSun

Mpc3 ∼ 10−30 gcm3 ∼ 10−6 GeV

cm3 , (4.15)

i.e., in the Universe is almost empty (∼ one Hydrogen atom per cubic metre).4Different energy components are expressed as ratios with respect to 𝜌crit,

that can be thought of, in a hand-waving fashion, as a measure of the differentweights of kinetic 𝐾𝑖 ∼ 1

2𝑚𝑖��2 and potential energy 𝑈𝑖 ∼ 𝐺𝑀𝑚𝑖

𝑎for a particle

of mass 𝑚𝑖, as can be trivially seen using 𝑉 ∼ 43𝜋𝑎

3:

Ω𝑖 ≡ 𝜌𝑖𝜌crit

∼ 𝑈𝑖𝐾𝑖

. (4.16)

Using Ωtot = ∑𝑖 Ω𝑖, eq. 4.10 can be expressed in the following compact

and nice form:Ωtot − 1 = 𝑘

𝑎2 𝐻2 . (4.17)

4.3 Hubble’s Law and other cosmological pa-rameters

Let us introduce a few important cosmological concepts and parameters.Redshift is defined as

𝑧 ≡ 𝜆0 − 𝜆𝑒𝜆𝑒

= 𝑎0

𝑎𝑒− 1 , (4.18)

3The subscript 0 always denotes current values. Sometimes ℎ is used, which is today’sHubble constant, 𝐻0, in units of 100 km s−1 Mpc−1. A useful quantity to keep in mind isℎ2 ≃ 0.5.

4Notice the overdensity at our location in the Milky Way (MW), where 𝜌local ≈0.3 GeV cm−3.

4.3 Hubble’s Law and other cosmological parameters 43

where 𝜆0 is the observed wavelength of the radiation at a time 𝑡0 and 𝜆𝑒 isthe wavelength emitted at a time 𝑡𝑒 by the source which is at a comovingcoordinate 𝑟. The convention is to take 𝑎0 = 1.

The length of the spatial geodesic at a fixed time, 𝑑𝑡 = 0, is the properdistance 𝑑𝑝. Along a radial geodesic 𝑑Ω = 0, so we get from eq. 4.2:

𝑑𝑝(𝑡) = 𝑎(𝑡) 𝑟. (4.19)

Differentiating with respect to time, we get:

𝑣𝑟 = 𝐻0 𝑑𝑝, (4.20)

where for simplicity we have taken the peculiar velocities of the galaxies to bezero, i.e., 𝑣𝑝 ≡ 𝑎�� = 0. Therefore galaxies expand with recessional velocities𝑣𝑟 proportional to their proper distances. This is known as Hubble’s Law andhas been tested to a great accuracy (for large values of redshift there aredeviations, see below).

The luminosity distance is defined as

𝑑2𝐿 = 𝐿

4𝜋𝐹 , (4.21)

where 𝐿 is the luminosity of the source and 𝐹 the flux received by the observer.In an expanding Universe one obtains from energy conservation that

the flux received at our position, whose photons are redshifted by a factor1 + 𝑧 ≡ 𝑎0/𝑎, is:

𝐹 = 𝐿

4𝜋𝑎20𝑟

2(1 + 𝑧)2 . (4.22)

So from eqs. 4.21 and 4.23, we get that:

𝑑𝐿 = 𝑑𝑝(1 + 𝑧) . (4.23)

The deceleration parameter is defined as:

𝑞 = −𝑎��

��2 . (4.24)

To connect observations (redshift, luminosity distances) and the cosmologicalparameters (𝐻0, 𝑞0) let us consider the path travelled by light in a null-geodesic.From eq. 4.2:

0 = 𝑑𝑠2 = −𝑑𝑡2 + 𝑎2

1 − 𝑘𝑟2𝑑𝑟2, (4.25)

and taking 𝑘 = 0, we can integrate it from a time 𝑡1 to the current time, 𝑡0:

𝑟 =∫ 𝑡0

𝑡1

𝑑𝑡

𝑎(𝑡) . (4.26)


For small enough redshifts, we can expand the scale factor in a Taylor seriesaround its actual value 𝑎0:

𝑎(𝑡1) = 𝑎0 + (��)0Δ𝑡+ 12(��)0(Δ𝑡)2 + . . . , (4.27)

where Δ𝑡 = 𝑡1 − 𝑡0. Then, eq. 4.26 becomes:

𝑟 = 𝑎−10

[−Δ𝑡+ 1

2𝐻0Δ𝑡2 + . . . .]. (4.28)

Substituting eq. 4.27 in eq. 4.18, we get:

11 + 𝑧

= 1 +𝐻0Δ𝑡− 12𝑞0𝐻

20 Δ𝑡2 + . . . , (4.29)

from which Δ𝑡 can be obtained by expanding in 𝑧:

Δ𝑡 = 𝐻−10

[−𝑧 +

(1 + 𝑞0

2

)𝑧2 + . . .

]. (4.30)

Replacing eq. 4.30 in eq. 4.28 it is straightforward to obtain

𝑟 = 1𝑎0𝐻0

[𝑧 − 1

2 (1 − 𝑞0) 𝑧2 + . . .]. (4.31)

Finally, using eq. 4.23, we get:

𝑑𝐿 = 𝐻−10

[𝑧 + 1

2(1 + 𝑞0)𝑧2 + . . .]. (4.32)

For 𝑧 ≪ 1, we have that 𝑧 = 𝐻0 𝑑𝐿. Using that for low enough redshiftsthe recessional velocity is 𝑣𝑟 ≈ 𝑐𝑧 and 𝑑𝐿 ≈ 𝑑𝑝, we get Hubble’s Law. Atlarger redshifts we will have deviations from this law that are connected withthe deceleration parameter, as can be seen in eq. 4.32.

4.4 The energy content of the UniverseIn 1998, by using type Ia Supernovae, which are standard candles, i.e., theirluminosity profiles (𝐿) are known, measurements of 𝑑𝐿 (see eq. 4.21) showedthat the expansion of the Universe in recent epochs is accelerating [15, 16]. Acosmological constant Λ could provide the repulsive force needed to acceleratethe expansion, with 𝜌Λ = −𝑝 = Λ

8𝜋𝐺 , but this does not have to be the case,and in general we refer to this mysterious component with negative pressure asdark energy (DE). From these Supernovae luminosity distances, a combination

4.4 The energy content of the Universe 45

Figure 4.1: The temperature fluctuations of the CMB observed by Planck [85].

Figure 4.2: The energy content of the Universe by Planck [85].


of the total energy density in matter, Ωmatter, and that in dark energy, ΩDE,can be inferred.5

We show in figure 4.1 the CMB temperature fluctuations as a function ofthe angular scale [85]. Depending on the content of baryons, DM and DE, theshape, position and height of the CMB peaks vary. For instance, the positionof the first peak of the CMB tells us that the Universe is flat, Ωtotal = 1.Using latest Planck results [85], we know the following facts about the energycontent of the Universe today, illustrated in figure 4.2:

∙ It is dominated by DE, with ΩDE ∼ 0.68. If the current acceleration isunderstood as being caused by a cosmological constant Λ, which hasbeen negligible throughout the entire history of the Universe, this posesthe question of why precisely now Ωmatter ∼ ΩΛ.

∙ DM constitutes a significant fraction of the energy density in the Uni-verse, ΩDM ∼ 0.27.

∙ Visible matter, which is described by the first family of the SM (atoms)(see chapter 1) represents just a tiny fraction of the total, Ωb ≈ 0.05.6

∙ The photon and the neutrino energy densities are almost negligible.

So we have that:

Ωtotal = Ωb + ΩDM + ΩDE = 1. (4.33)

The baryon density Ωb has also been measured and agrees well with thesynthesis of elements in the Early Universe (a few seconds after the Big Bang),known as Big Bang Nucleosynthesis (BBN) [138–142]. At temperatures ∼ 1MeV, the light elements were produced with mass abundances of ∼ 75%(1H), ∼ 25% 4

2He, ∼ 0.01% deuterium (21H) and 3

2He, and ∼ 10−10% 73Li.

For instance, a somewhat hand-waving computation of the Helium massabundance is rather straightforward. Calling 𝑛𝑛 (𝑛𝑝) the number density ofneutrons (protons), roughly the final number of He nuclei is 1/2𝑛𝑛, as twoneutrons per He atom are needed. The mass is 𝑚He ∼ 4𝑚𝑛, so 𝜌He ≈ 2𝑛𝑛𝑚𝑛

and the total mass is 𝜌tot ≈ (𝑛𝑛 + 𝑛𝑝)𝑚𝑛, where we took 𝑚𝑛 ∼ 𝑚𝑝. The finalmass fraction of Helium 𝑌He is therefore given by:

𝑌He ≡ 𝜌He

𝜌tot= 2(𝑛𝑛/𝑛𝑝)

1 + (𝑛𝑛/𝑛𝑝), (4.34)

5Λ can be also interpreted as entering on the left-hand side of the GR equations, eq.4.3, forming part of the geometry of the Universe, although it is probably easier to think ofit as an extra energy component.

6There is no better lesson in humility towards our knowledge of the Universe as thispiece of data.

4.4 The energy content of the Universe 47

which just depends on the neutron to proton fraction at the moment ofNucleosynthesis, given by their equilibrium distributions:

𝑛𝑛𝑛𝑝

∼ 𝑒−(𝑚𝑛−𝑚𝑝)

𝑇Nucl . (4.35)

Neutrons interact with positrons or electron neutrinos to create protons andother products, until the expansion of the Universe is larger than these rates,moment at which they freeze-out. If Nucleosynthesis occurred very early,𝑇Nucl ≫ 10 MeV, then 𝑛𝑛/𝑛𝑝 ∼ 1 and 𝑌He → 1. If too late, 𝑇Nucl ≪ 1 MeV,then 𝑛𝑛/𝑛𝑝 ∼ 0 and 𝑌He → 0. At 𝑇Nucl ∼ 0.7 MeV which is the correct freeze-out temperature, 𝑛𝑛/𝑛𝑝 ∼ 1/67, and at 𝑇Nucl ∼ 0.1 MeV, when the Universehad cooled sufficiently to allow deuterium nuclei to survive disruption by high-energy photons, 𝑛𝑛/𝑛𝑝 ∼ 1/7, for which one gets 𝑌He ∼ 0.25, in agreementwith observations.

The excellent agreement between the theoretical computations and theobservations is one of the main evidences of a Hot Big Bang, together withthe observation of the CMB.

Analysis of Large Scale Structure (LSS), like the study of the photon-baryon fluid before recombination, where the competition between gravityand radiation pressure produces oscillations in the plasma which propagate asacoustic waves, known as Baryon Acoustic Oscillations (BAO), can be usedto infer the cosmic expansion. Also from observations of clusters of galaxies,for instance, via gravitational lensing, one gets information on Ωmatter.

For a mass density 𝜌(r), one can define the density contrast 𝛿(r) ≡(𝜌(r)−𝜌0)/𝜌0, where 𝜌0 is the mean value of the matter density. The matterpower spectrum, 𝑃 (k), is just the Fourier transform of the correlation functionof the matter density contrast, 𝜉(r) = ⟨𝛿(r1)𝛿(r1 + r)⟩, i.e.:

𝑃 (k) =∫𝜉(r) ei k·r d3r, (4.36)

and can be reconstructed from different observations, as shown in figure 4.3.As will be discussed in chapter 5, if DM was hot, 𝑃 (k) at low scales wouldbe suppressed (also warm dark matter modifies it, but it is compatible).

The different cosmological observations from CMB, BAO and Supenovaeagree very well, as can be seen in figure 4.4, and it is sometimes known as theConcordance Model (another name of the ΛCDM model).

7This fraction is not constant, as the neutron decays.


Figure 4.3: The matter power spectrum 𝑃 (k) reconstructed from differentobservations [143].

4.5 An inflationary epochThe evidence for an inflationary epoch in the first stages of the Universe,at 𝑡 ∼ 10−34 s, is very strong. Looking at eq. 4.11, we see that to have anaccelerated expansion, �� > 0, we need that, if the component 𝑖 dominates theenergy density, 𝜌𝑖(𝑡) + 3𝑝𝑖 < 0 (i.e., 𝜔𝑖 < −1/3).

This is equivalent to demanding that the comoving Hubble radius decreaseswith time:

𝑑

𝑑𝑡

( 1𝑎𝐻

)≡ −��𝑎𝐻

< 0. (4.37)

One of the most important pieces of evidence for an epoch of acceleratedexpansion comes from the CMB. As shown in figure 4.1, where the CMBtemperature fluctuations are plotted as a function of the angular scale [85], theCMB is extremely homogeneous, with temperature fluctuations 𝛿𝑇/𝑇 ∼ 10−5.The problem is that within the standard Big Bang paradigm, these regionswere never causally connected. This is known as the horizon problem and itis easy to show how inflation solves it by looking at figure 4.5 (from [34]),which shows the evolution of the comoving Hubble radius, 1/(𝑎𝐻(𝑡)), whichis the maximum distance between causally connected regions (the maximum

4.5 An inflationary epoch 49

Figure 4.4: The concordance within the ΛCDM model of the differentcosmological observations coming from the CMB (Planck and WP), BAO(SDSS-II, BOSS and 6dFGS) and Supernovae (Union2), from [144]. Dark andlight regions indicate 68.3% and 95.4% CL.

Figure 4.5: Left: The comoving Hubble sphere shrinks during inflation andexpands after inflation. Right: all scales relevant to cosmological observationstoday were larger than the comoving Hubble radius until recently (came backwithin the Hubble radius at relatively recent times). However, at sufficientlyearly times, these scales were smaller than the Hubble radius and thereforecausally connected.


distance light has had time to travel), shrinking during inflation and expandingafterwards.

All scales that are relevant to cosmological observations today were largerthan the Hubble radius until recently - having reentered the Hubble radius atrelatively recent times. However, at sufficiently early times, these scales weresmaller than the Hubble radius and therefore causally connected if inflationhappened.

As has been already said, current observations of the CMB by the Plancksatellite [85] tell us that the Universe is flat, i.e., 𝑘 = 0 in eq. 4.2, andtherefore Ωtotal ≡ 𝜌total

𝜌crit= 1.8 In the standard picture, the right-hand side of

eq. 4.17 always grows with time (∝ 𝑡 in RD (∝ 𝑡2/3 in MD)), and therefore Ωwould have to be severely fine-tuned to one in the past to be observed todayas having a value equal to one. This is known as the flatness problem, and aperiod of accelerated expansion is able to solve it, as can be clearly deducedfrom eq. 4.37. Last but not least, the density fluctuations that we see in theCMB, from which structures grew, are caused by the primordial fluctuationsof the inflationary field Φ and the metric 𝑔𝜇𝜈 (Φ(��, 𝑡) ≡ Φ(𝑡) + 𝛿Φ(��, 𝑡),𝑔𝜇𝜈(��, 𝑡) ≡ 𝑔𝜇𝜈(𝑡) + 𝛿𝑔𝜇𝜈(��, 𝑡)).

Figure 4.6: Contours showing the 68% and 95% CL regions for 𝑟 and thescalar spectral index 𝑛𝑠, from [12].

Inflation implies that the quantization of the gravitational field coupledto the exponential expansion produces a primordial background of stochastic

8The first acoustic peak determines the size of the horizon at the time of decoupling,and therefore the geometry of the Universe.

4.5 An inflationary epoch 51

gravitational waves, that would give a unique signature on the CMB vialocal quadrupole anisotropies, that include a curl or B-mode component atdegree angular scales, which depends on the tensor-to-scalar ratio, 𝑟. Recently,the first detection of primordial B-modes was made by BICEP2 [12], in therange 30 < 𝑙 < 150, inconsistent with the null hypothesis at several standarddeviations. They give a tensor-to-scalar ratio of 𝑟 = 0.20+0.07

−0.05. This ratio inthe simplest models tells us that the typical energy scale of inflation is similarto the GUT scale, ∼ 1016 GeV. In figure 4.6 from [12] are shown contoursin the tensor (𝑟) and the scalar spectral index (𝑛𝑠) plane. As scalars do notproduce B-modes while tensors do, the detection of B-modes is a smoking gunof tensor modes, and therefore of inflation. This discovery, which as always inphysics should be confirmed by other independent experiments, like Planck,is to date the most compelling evidence of a nearly exponential expansionwhich sets the initial conditions for the subsequent Hot Big Bang, understoodas the creation of particles from inflaton decays, not as the initial singularity.

For different inflationary models and a more extense discussion, the readeris referred to [34], a nice review of the topic.

Chapter 5

Dark matter

In this chapter we discuss the evidence and properties of DM. We willthen focus on Weakly Interacting Massive Particles (WIMPs) and detectionprospects via indirect searches, colliders and cosmological imprints. We leavedirect detection, which is the focus of this thesis, to chapter 6.

5.1 Evidence of dark matterThere is plenty of evidence for the existence of some extra matter in theUniverse, whose properties we will detail in the next section. The mostimportant pieces of evidence are the following:

1. Rotation curves of spiral galaxies.Like the planets rotating around the Sun, one would expect that lumi-nous matter in the galaxies, basically Hydrogen, should have a rotationspeed that would decrease with the distance to the galactic centreas 1/

√𝑟.1 However, one observes2 that the rotation velocity remains

practically constant (𝑣𝑟𝑜𝑡 ∼ 𝑣𝑆𝑢𝑛 ∼ 220 km/s) up to large distances, seefigure 5.1. This can not be understood with just the luminous matter,which decreases exponentially with the distance: there must be a muchlarger gravitational potential than that created by luminous matter.There are non-DM possibilities to explain this piece of evidence, forinstance if we modify gravity at these scales (Modified NewtonianDynamics, MOND [145,146] or its relativistic version [147]). Its basic

1In reality there are two components of luminous matter with different behaviours, thebulge and the disc, as can be seen looking at figure 5.3.

2In the Milky Way (MW) we can measure its rotation curve using the Hydrogen 21 cmline, as there is too much dust to observe X-rays.

54 Chapter 5. Dark matter

assumption is that Newton’s Second Law has only been tested at largeaccelerations. The new modified equation would be:

𝐹 = 𝑚𝜇[𝑎/𝑎0] 𝑎, (5.1)

where 𝜇 = 1 for 𝑎 ≫ 𝑎0, recovering Newton’s Second Law, and 𝜇 = 𝑎/𝑎0if 𝑎 ≪ 𝑎0. Therefore, for a star of mass 𝑚 far away from the galaxy ofmass 𝑀 at a distance 𝑟, we have:

𝐹 = 𝐺𝑀𝑚

𝑟2 = 𝑚𝜇[𝑎/𝑎0] 𝑎, (5.2)

and we get for 𝑎 ≪ 𝑎0:

𝑎 =√𝐺𝑀𝑎0

𝑟→ 𝑣 = (𝐺𝑀𝑎0)1/4 = constant. (5.3)

The only parameter of the theory, 𝑎0, is similar for all galaxies and hasa curious value, 𝑎0 ∼ 1.2 · 10−10 m s−2 ∼ 𝑐𝐻0.

Figure 5.1: Rotation velocity as a function of distance. It can be seen howthe sum of the disc and the halo components give the observed final constantspeed.

2. Mass-to-light ratio of galaxy clusters.

5.1 Evidence of dark matter 55

One can apply the Virial Theorem (2𝑇 + 𝑉 = 0) to galaxy clusters,which are gravitationally bounded systems. From X-rays one gets thetemperature of the baryons in the gas, and therefore their velocities,from which the mass of the cluster can be inferred. In the 1930’s, F.Zwicky applied this procedure to the galaxies inside the Coma Cluster,and obtained a very large mass-to-light ratio of ∼ 400 (in solar units)3:more matter than the luminous one is present.

3. Collisions of clusters, like the Bullet Cluster.In figure 5.2 [148] we see the result of the merging of two galaxy clusters.Via gravitational lensing we can infer that most of the matter of thecluster is located in the blue regions, while from X-rays observationswe see that the luminous matter, basically the gas of the clusters thatcollided, is located in the red regions. It is very difficult to conceivehow gravity in a modified theory will point to regions (the blue ones)where no matter is present.

Figure 5.2: Picture of the Bullet Cluster. In blue the matter component,inferred from gravitational lensing; in red, the luminous component, fromX-rays.

4. The anisotropies of the Cosmic Microwave Background (CMB).As was discussed in chapter 4, by measuring the positions and relativeheights of the acoustic peaks of the CMB [85,149], see figure 4.1, one candeduce the energy content of the Universe. There are two competingforces in the photon-baryon-dark matter plasma: radiation pressure thatacts as a repulsive force, and gravity that is an attractive force. DM

3Currently typical values of the total matter density in clusters are Ω𝑀 ∼ 0.2 − 0.3.


does not couple to photons, while baryons do, and therefore the relativeheights of the peaks give us information about the relative amount ofDM (mainly the third peak) compared to baryonic matter (mainly thesecond peak).

5. The growth of structure.From N-body simulations we infer that the growth of structures inthe Universe is a hierarchical process. During radiation dominationthe growth is slow; clustering becomes more efficient when matterdominates. As DM does not dissipate energy, it seeds large potentialwells, where baryonic matter becomes gravitationally trapped, formingtherefore small scales that are gravitationally bound and decouple fromthe overall expansion. This leads to a picture of hierarchical structureformation with small-scale structures (like stars and galaxies) formingfirst and then merging into larger structures (clusters and superclustersof galaxies).

6. Globular Clusters.These are regions of ∼ 105 stars, very old, that would escape the galaxy,as they are moving very fast, if the galactic potential well was just theone created by the luminous matter.

7. Other cosmological observations, like the matter power spectrum, BBNand gravitational lensing tell us that the amount of matter is notcompatible with the luminous one.

5.2 Properties of dark matterAlthough there are exceptions, the main properties that DM particle 𝜒 shouldhave are the following:

∙ It interacts gravitationally.

∙ Collision-less, it does not dissipate. This is the main reason why DMdoes not lose energy and therefore forms ellipsoidal haloes (see figure5.3 for our halo in the Milky Way), unlike baryonic matter, that formsdiscs due to energy losses and angular momentum conservation.

∙ Cold (or warm) at the time of decoupling, i.e., 𝑚𝜒 > 𝑇dec, otherwise itwould have free-streamed erasing the smaller structures. The matterpower spectrum 𝑃 (k), as shown in figure 4.3, gives us very valuable

5.2 Properties of dark matter 57

information about the type of DM particle. For typical CDM, thesmaller scales that can be formed are ∼ 10−6 𝑀𝑆, while for typicalWDM (HDM) these are ∼ 109 𝑀𝑆 [150] (∼ 1015 𝑀𝑆), yielding for thema non-hierarchical growth of structure. This rules-out HDM as a primarycomponent, as 𝑃 (k) would be suppressed at the smaller scales, whileWDM is still allowed, even as the dominant component.

∙ Neutral (no tree level EW interactions). It does not emit or absorb light.Otherwise the CMB as well as the matter spectrum would be different.In addition, bounds on charged DM (CHAMPs) [151] from looking forheavy isotopes or heavy water are very strong, see for instance [152].

∙ Colorless (no tree level color interactions). In general, DM that isstrongly interacting with nucleons is severely constrained, for instancefrom measurements of the Earth’s heat flow [153].

∙ Long-lived with respect to the age of the Universe (or stable), so it ispresent today with the observed abundance. Light DM candidates, suchas the axion or the sterile neutrino, are naturally long-lived as theirlifetimes are inversely proportional to the mass to the fifth power. Forthe case of WIMPs, however, some symmetry is usually needed.

∙ It may interact weakly with the SM. If it is a thermal relic, in principleit is hoped that it talks to the SM, see section 5.3, although this doesnot have to be the case.

∙ It has to be a new particle not present in the SM: baryonic objectssuch as interstellar gas, massive compact halo objects (MACHOs) orprimordial black holes [154] are ruled-out as a significant component.MACHOs are ruled-out by gravitational lensing searches [155] and blackholes in several mass ranges are severy constrained, see for instance [156].In addition, BBN imposes severe constraints on the amount of lightelements, which matches the amount of gas (basically Hydrogen andHelium) observed [157]. Neutrinos (HDM) are also ruled-out as theprimary component: we know that relic neutrinos are part of the DMenergy density, but not a significant fraction as they are hot. Well-motivated non-baryonic candidates include axions (can explain theCP problem of the SM) or neutralinos (SUSY solves theoretically theHierarchy Problem).4

4For more information on WIMP candidates (among other DM topics, such as theevidence and properties of DM) the reader is referred to the review [158]. For a minimalisticapproach, see [159], where the different multiplets with their assignments that can be added


∙ It can have whatever spin: 0, 1/2, 1...

∙ There could be more than one non-baryonic DM particle, i.e., contribut-ing significantly to the DM energy density of the Universe. Why wouldthere just be one stable DM? In the SM there are a few stable (orextremely long-lived) particles: electrons (lightest electrically chargedparticle), protons (lightest particle carrying baryon number, whichcould decay only by higher-dimensional operators), photons (massless,by kinematics), gluons (lightest colored states) and the lightest fermion,which is the lightest neutrino, stable thanks to angular momentumconservation.

Figure 5.3: Pictorical view of the Milky Way (MW), with the disc, the bulgeand the DM halo. Figure taken from the lecture notes "Galaxies", by PhilUttley.

to the SM to provide a neutral stable particle, that is a viable DM candidate, are studied(like a spin 1/2 𝑌 = 0 𝑆𝑈(2)L quintuplet which has a stable particle).

5.3 Thermal freeze-out and the WIMP paradigm 59

5.3 Thermal freeze-out and the WIMP paradigmIn the Hot Big Bang paradigm, basic ingredient of the ΛCDM model, theEarly Universe is a hot and dense soup of particles in thermal equilibrium. Theparticle interaction rate, Γ ∼ 𝑛 ⟨𝜎𝑣⟩, is initially, when the temperature is veryhigh, much larger than the Hubble expansion rate, 𝐻. Thermal decouplinghappens at 𝑇 = 𝑇f.o., when the particle interaction rate becomes of the sameorder of the Hubble expansion rate, Γ(𝑇f.o.) ∼ 𝐻(𝑇f.o.). To compute relicdensities one should solve Boltzmann equations, that describe the evolutionof the number densities, 𝑛(𝑡):

𝑑𝑛

𝑑𝑡+ 3𝐻𝑛 = −⟨𝜎𝑎𝑛𝑛𝑣⟩[(𝑛)2 − (𝑛𝑒𝑞)2], (5.4)

where 𝑛𝑒𝑞 is the equilibrium number density. However, here we will justdiscuss what happens qualitatively.

If this freeze-out (f.o.) occurs in the radiation-dominated era, 𝑇f.o. >𝑇M−R equality ∼ 1 eV, we can use that (𝑛hot ∝ 𝑇 3)

𝜌 ∼ 𝜌rad = 𝜋2

30 𝑔* 𝑇4, (5.5)

with 𝑔* the number of relativistic degrees of freedom at 𝑇f.o.. Therefore, usingeq. 4.10 (setting 𝑘 = 0), we can approximate:

𝐻 =(8𝜋𝐺

3 𝜌)1/2

≃ 𝑇 2

𝑀𝑝

, (5.6)

where we used that 𝑀𝑝 =√

18𝜋𝐺 .

Depending on whether the particle decoupling happens while being rela-tivistic (𝑇f.o. > 𝑚𝜒) or non-relativistic (𝑇f.o. < 𝑚𝜒), it is called a hot or coldrelic.

Neutrinos, that are non-relativistic today (at least two of them)5, but arerelativistic (hot) when they decouple at 𝑇 𝜈f.o. ≃ 1 MeV, have a relic densitygiven by:6

Ω𝜈ℎ2 ≃

∑𝑖𝑚𝜈𝑖

93 eV . 10−2, (5.7)

where we have used the approximate upper bound on the sum of neutrinomasses coming from different cosmological observations, ∑𝑚𝜈𝑖 . 1 eV. As

5If there is a relativistic neutrino today, its contribution to the energy density isΩ𝜈ℎ2 ∼ 5.6 · 10−5.

6Before knowing the upper bound on neutrino masses that comes from cosmology, forinstance from not erasing the smaller scales of the Universe, one could set a limit on∑

𝑚𝜈𝑖 . 𝒪(20) eV so the Universe was not over-closed.


we know from neutrino oscillations (see chapter 2) that ∑𝑚𝜈𝑖 & 0.06 (0.1) eVfor NH (IH), there are also lower bounds for their contribution to the energydensity: Ω𝜈ℎ

2 & 6 · 10−4 (∼ 10−3).WIMPs are DM particles with weak scale masses, 𝑚𝜒 = 𝒪(1 − 103) GeV,

and interactions, 𝜎𝜒 = 𝒪(10−8) GeV−2. They are among the most popularcandidates (several examples exist, such as the neutralino in Supersymmetry),and in principle they can be detected.

When the temperature of the Universe was larger than their mass, 𝑇 > 𝑚𝜒,the number densities of photons and WIMPs were similar and 𝑛𝜒 ∼ 𝑛𝛾 ∝ 𝑇 3

and WIMPs annihilated into lighter particles and viceversa. But when thetemperature dropped below their mass, their number density was reducedexponentially, 𝑛𝜒 ∝ 𝑒

−𝑚𝜒𝑇 . As a consequence, Γ < 𝐻 and they froze-out.

With their weak scale values, the final relic abundance is found to be roughlythe observed one. This is called the WIMP miracle, as the final abundance isproportional to the one at the equilibrium, which for CDM (non-relativistic)depends exponentially on the temperature at freeze-out 𝑇f.o. and on the mass(it is Boltzmann suppressed). The freeze-out temperature 𝑇f.o. depends onthe DM mass and only logarithmically on the cross-section..., yielding a finalvalue of 𝑚𝜒/𝑇f.o. ≈ 20 − 25, quite insensitive to the particular kind of theDM interactions and masses.

Neglecting decays of the DM particles7, the final expression for the currentnumber density is:

𝑛today = 𝑛fo = 𝐻f.o.

⟨𝜎ann𝑣⟩ f.o., (5.8)

and the relic density can be easily computed to be:

ΩDMℎ2 ∝ 3 × 10−27 cm3 s−1

⟨𝜎ann𝑣⟩f.o., (5.9)

where one can see that for weak scale couplings it is easy to achieve theobserved value, ΩDMℎ

2 ∼ 0.1.There is an allowed range in masses for CDM relics: 𝒪(10)GeV ≤ 𝑚𝜒 ≤

𝒪(100)TeV. The upper bound comes from the fact that couplings cannotbe arbitrarily large, while the lower bound, known as the Lee-Weinberglimit [160], comes from not over-closing the Universe for weak scale cross-section (𝜎 ∼ 𝐺2

𝐹𝑚2𝜒) DM particles.

Exceptions of this typical thermal freeze-out exist [161], the most relevantones being the existence of resonances (with mass 𝑚𝑅 ≃ 2𝑚𝜒), thresholdeffects (which imply that 𝜎f.o. > 𝜎today) and co-annihilations (for various DMparticles with similar masses, 𝑚heavy −𝑚light . 𝑇f.o.).

7There could be entropy injections from relic decays, for instance, which would dilutethe final densities.

5.4 Asymmetric dark matter 61

5.4 Asymmetric dark matterIn the standard paradigm for DM via thermal freeze-out, the DM and baryonrelic densities arise from completely different mechanisms, and the fact thattheir densities are of the same order of magnitude is a coincidence. AsymmetricDM (see for instance [162–164], [165] for a nice implementation and [166] fora recent review on the topic) proposes that, similarly to baryons, a particle-antiparticle asymmetry of DM particles is left over, linked to the ordinarybaryonic one. Therefore, the number densities are similar, 𝑛𝜒 ∼ 𝑛baryons, andthe masses fulfill 𝑚𝜒 ∼ 5𝑚baryons ∼ 5 GeV.

The two basic ingredients an asymmetric DM model needs to have are:

∙ a conserved (or approximately-conserved) dark global quantum number,so that a dark asymmetry exists.

∙ an interaction that erases the symmetric part, just as in the SM thestrong and electroweak interactions annihilate the symmetric componentof the SM into photons.

Apart from these, the new dark sector could be as complicated (or more) asthe SM.

In addition to being theoretically very appealing, asymmetric DM modelsrecently gain attention as an explanation for the low DM mass hints observedby some direct detection experiments, as will be shown in chapter 6. Aspecial feature is that indirect detection through DM particle-antiparticleannihilations is absent, because there are no DM antiparticles left to annihilatewith, so direct detection seems in principle a more promising way to test thishypothesis.

5.5 Dark matter searchesIn this thesis we focus on direct detection, therefore we next briefly review thestatus of other interesting ways to search for DM: indirect detection, collidersearches and via cosmological measurements, and leave direct detection tochapter 6. These searches assume (hope) that DM interacts at some pointwith the SM.

5.5.1 Dark matter indirect detectionIn principle, one could observe the annihilation and decay products of DMparticles: neutrinos, photons (gamma rays, X-rays, synchrotron radiation),


electrons and positrons, protons and antiprotons, anti-deuterons (bound statesof 𝑝 and ��)...

These two different processes are given in general by the following expres-sions:

∙ Particle decay:

Γdecay =∫𝑛𝜒 𝑑𝑉 · 1

𝜏decay·𝑁decay

SM . (5.10)

∙ Particle annihilation:

Γann =∫𝑛2𝜒 𝑑𝑉 · 𝜎ann 𝑣 ·𝑁ann

SM . (5.11)

In both cases the first term is the number of particles in the volume 𝑉 , thesecond is the decay/annihilation rate and the third is the flux of SM particlesper event.

WIMPs are therefore expected to annihilate very efficiently in regions withhigh densities, such as the galactic centre, galaxy clusters, dwarf spheroidals,the Sun... Of course, separating astrophysical origins (cosmic rays) from trueDM particle decays is very challenging, and therefore anti-particles ratherthan particles, and heavy nuclei, which have much smaller backgrounds, arecommonly searched for.

One of the smoking guns of DM would be for instance high energy neutrinospointing directly to the Sun or a gamma ray line coming from the galacticcentre, as they are not deflected by the magnetic fields of the galaxy.

The experimental situation is controversial, and although there is noconfirmed DM discovery, there have been several claims in the past years.For example, Pamela, ATIC and Fermi have shown an excess with respect tothe background of positrons in the ∼tens to ∼hundreds of GeV.

A DM explanation of the PAMELA excess faces the problem that noantiproton excess is observed (and no excess of radio emission or gammarays). In addition, the DM particle should have an annihilation cross-sectionmuch larger than the one needed to explain the observed abundance.8 All inall, pulsars seem to be a plausible explanation for this excess.

8The annihilation cross-section, 𝜎ann, does not have to be the same at freeze-out andat current times. If it is dominated by the p-wave contribution, it may happen that𝜎f.o.

ann > 𝜎nowann . Therefore, the 𝜎now

ann extracted from absence of indirect signals constitutes alower bound for 𝜎f.o.

ann.

5.5 Dark matter searches 63

5.5.2 Collider searchesDM particles, being very-weakly interacting and stable, are expected to escapedetection at colliders, yielding large amounts of missing energy and otherparticles (for instance a jet or a photon) [167–169]. The limits set by thesekind of searches at the Tevatron and LHC are competitive with those comingfrom direct searches, especially for spin-dependent interactions and light DMmasses.

An important caveat to discover DM at colliders is that one can never saythat they are stable enough, nor that their annihilation cross-section is suchthat, they are present today with the correct relic abundance and thereforeconstitute the DM of the Universe. Therefore, even if a stable neutral particleis observed at a collider, direct and indirect detection of it are needed beforeclaiming it is the DM.

5.5.3 Imprints in BBN and in the CMBDM annihilations and decays inject energy and particles, changing the historyand properties of the Universe at different epochs. These energy injectionscan change the formation of light nuclei (BBN) (see for instance [170,171]) orthe anisotropies of the CMB (see for instance [172, 173] and more recently[174,175]), yielding very strong constraints on the annihilation cross-section,especially in the case of the CMB.

A word of caution is in order in all indirect searches: in almost everyanalysis, the rates (and therefore the final constraints) depend crucially onthe channel, and are therefore model-dependent.

Chapter 6

Direct detection of dark matter

We know that DM interacts gravitationally. A fundamental question iswhether DM interacts also non-gravitationally. We review in this chapterdirect detection: experiments that are looking for the scattering of DMparticles from the galactic halo in underground detectors.

6.1 The event rateAs seen in chapter 5, particles that freeze-out with weak scale masses andcouplings, or Weakly Interacting Massive Particles (WIMPs), are among themost interesting candidates, and at some level we expect that they interactwith the SM.1 If DM is a WIMP it may induce an observable signal inunderground detectors by depositing a tiny amount of energy after scatteringwith a nucleus in the detector material [176]. Many experiments are currentlyexploring this possibility and delivering a wealth of data.

A useful picture to visualize direct detection is illustrated in figure 6.1,where one sees that, in our reference frame, as we move through the galactichalo, there is a DM flux of particles that goes through the Earth.

This flux gives rise to a differential event rate. The rate for a DM particle𝜒 to scatter off a nucleus (𝐴,𝑍) and deposit the nuclear recoil energy 𝐸𝑛𝑟 inthe detector is given by 𝑅(𝐸𝑛𝑟, 𝑡):

𝑅(𝐸𝑛𝑟, 𝑡) = 𝜌𝜒𝑚𝜒

1𝑚𝐴

∫𝑣>𝑣𝑚

𝑑3𝑣𝑑𝜎𝐴𝑑𝐸𝑛𝑟

𝑣𝑓det(v, 𝑡), (6.1)

and it is measured in events/keV/kg/day. 𝑚𝐴 and 𝑚𝜒 are the nucleus andDM masses, 𝜎𝐴 the DM–nucleus scattering cross section and v the 3-vector

1Really, we just hope that they do interact with the SM, as there could be just anindependent dark sector with no interactions with the SM.

66 Chapter 6. Direct detection of dark matter

Figure 6.1: The WIMP flow in our rest-frame.

relative velocity between DM and the nucleus, with 𝑣 ≡ |v|.In direct detection, only the DM density at our location in the MW

matters, 𝜌𝜒. It has some uncertainty, but it affects all experiments by a globalfactor. Typical values are 𝜌𝜒 ≃ 0.3 GeV/cm3. There are many different DMdensity profiles of our galaxy in the literature, one of the most popular onesbeing the Navarro-Frenk-White (NFW) (see [177] for a discussion on CDMhaloes):

𝜌(𝑟) ∝ 1𝑟(1 + 𝑟/𝑟𝑠)2 , (6.2)

with 𝑟𝑠 ∼ 20 kpc.𝑓det(v, 𝑡) describes the distribution of DM particle velocities in the detector

rest frame, with 𝑓det(v, 𝑡) ≥ 0 and∫𝑑3𝑣𝑓det(v, 𝑡) = 1, and it is related to the

velocity distribution in the rest frame of the Sun, 𝑓sun and at the galaxy, 𝑓gal,by:

𝑓det(v, 𝑡) = 𝑓sun(v + v𝑒(𝑡)) = 𝑓gal(v + v𝑒(𝑡) + vsun), (6.3)

with vsun ≈ (0, 220, 0) km/s + vpec and vpec ≈ (10, 13, 7) km/s the peculiarvelocity of the Sun. We are using galactic coordinates where 𝑥 points towardsthe galactic center, 𝑦 in the direction of the galactic rotation, and 𝑧 towardsthe galactic north, perpendicular to the disc.

The velocity vector of the Earth, v𝑒(𝑡), is relevant for the annual mod-ulation signal, which will be described in the following section, and can bewritten as [178]

v𝑒(𝑡) = 𝑣𝑒[e1 sin 𝜆(𝑡) − e2, cos𝜆(𝑡)] (6.4)

with 𝑣𝑒 = 29.8 km/s, and 𝜆(𝑡) = 2𝜋(𝑡 − 0.218) with 𝑡 in units of 1 yearand 𝑡 = 0 at January 1st, while e1 = (−0.0670, 0.4927,−0.8676) and e2 =(−0.9931,−0.1170, 0.01032) are orthogonal unit vectors spanning the plane ofthe Earth’s orbit, assumed to be circular. As shown in [179], eq. (6.4) provides

6.1 The event rate 67

an excellent approximation to describe the annual modulation signal. Formore information on the relevance of the Earth’s velocity for direct detectionexperiments we recommend the recent analysis [180].

The integral of the velocity distribution enters in 𝑅(𝐸𝑛𝑟, 𝑡) because DMscattering at different angles probes different DM velocities even for fixed 𝐸𝑛𝑟.For a DM particle to deposit a recoil energy 𝐸𝑛𝑟 in the detector, a minimalvelocity 𝑣𝑚 is required by kinematics, restricting the integral over velocitiesin eq. (6.1).

For inelastic scattering between two DM particles close in mass, we havethat the minimal velocity required for a recoil of energy 𝐸𝑛𝑟 is:

𝑣𝑚 =√

12𝑚𝐴𝐸𝑛𝑟

(𝑚𝐴𝐸𝑛𝑟𝜇𝜒𝐴

+ 𝛿

), (6.5)

where 𝜇𝜒𝐴 is the reduced mass of the DM-nucleus system, and 𝛿 is themass splitting between the two DM states. For each value of 𝐸𝑛𝑟 there is acorresponding 𝑣𝑚 while the converse is not always true: certain values of 𝑣𝑚correspond to two values of 𝐸𝑛𝑟. We study inelastic scattering in detail in [6],where this simple observation gave rise to shape test explained there.

For elastic scattering, 𝛿 = 0, eq. 6.5 reduces to

𝑣𝑚 =⎯⎸⎸⎷𝑚𝐴𝐸𝑛𝑟

2𝜇2𝜒𝐴

. (6.6)

It is interesting to notice that for light DM particles, i.e., in the limit𝑚𝜒 ≪ 𝑚𝐴,𝑣𝑚 ∝ 1/𝑚𝜒, and therefore we are probing the tail of the velocity distribution.In principle, the integral is extended up to arbitrarily large velocities, however,there is a maximum speed that DM particles can have to be gravitationallybound to the galaxy, known as the escape velocity, 𝑣esc (more on it later).

The particle physics enters in eq. (6.1) through the differential cross section.For direct detection to make sense, DM should couple at some level to upand/or down quarks and/or electrons. Of course, it could happen that DMinteracts with these first-family fermions via loops.

For the standard spin-independent (SI) and spin-dependent (SD) scatteringthe differential cross section has the form

𝑑𝜎𝐴𝑑𝐸𝑛𝑟

= 𝑚𝐴

2𝜇2𝜒𝐴𝑣

2𝜎0𝐴𝐹

2(𝐸𝑛𝑟) , (6.7)

where 𝜎0𝐴 is the total DM–nucleus scattering cross section at zero momentum

transfer, and 𝐹 (𝐸𝑛𝑟) is the nuclear form factor, different for SI and SD. IfDM is composite, a DM form factor 𝐹𝐷𝑀(𝐸𝑛𝑟) appears here, but we neglectthis possibility for the rest of the thesis.


SI interactions arise from scalar and vector couplings to quarks, and 𝜎0𝐴

can be written for them as

𝜎SI𝐴 = 𝜎𝑝[𝑍 + (𝐴− 𝑍)(𝑓𝑛/𝑓𝑝)]2𝜇2

𝜒𝐴/𝜇2𝜒𝑝 , (6.8)

where 𝜎𝑝 is the DM–proton cross-section and 𝑓𝑛,𝑝 are coupling strengths toneutron and proton, respectively. If 𝑓𝑛 = 𝑓𝑝 the interactions are isospin-violating (IV).

SD interactions arise from axial-vector couplings to quarks, and 𝜎0𝐴 can

be written for them as

𝜎SD𝐴 =

32𝜇2𝜒𝐴

𝜋𝐺2𝐹

(𝐽 + 1)𝐽

(𝑎𝑝𝑆𝑝 + 𝑎𝑛𝑆𝑛)2, (6.9)

where 𝐽 is the spin of the nucleus, 𝑆𝑝 (𝑆𝑛) are the average spin contributionsfrom the proton (neutron) groups and 𝑎𝑝 (𝑎𝑛) are the effective couplings tothe proton (neutron).

In general both interactions contribute, although for nuclei with massnumber 𝐴 > 20 the SI contribution usually dominates.

It will be very useful for the rest of the thesis to define the halo integral

𝜂(𝑣𝑚, 𝑡) ≡∫𝑣>𝑣𝑚

𝑑3𝑣𝑓det(v, 𝑡)

𝑣, (6.10)

using which the event rate is given by

𝑅(𝐸𝑛𝑟, 𝑡) = 𝐶 𝐹 2(𝐸𝑛𝑟) 𝜂(𝑣𝑚, 𝑡) with 𝐶 = 𝜌𝜒𝜎0𝐴

2𝑚𝜒𝜇2𝜒𝐴

. (6.11)

The coefficient 𝐶 contains the particle physics dependence, while 𝜂(𝑣𝑚, 𝑡)encodes the astrophysics dependence.

6.2 The annual modulation signalThe direct detection experiments measure the nuclear recoils via light, chargeand/or heat. Some of them are capable of measuring two of these signals,which allow them to discriminate between electron (expected for backgrounds)and nuclear recoils (expected for DM interactions). In addition, DM willscatter at most once, and this is used to discriminate between nuclear recoilscoming from DM and those coming from, for instance, neutrons, which couldscatter more than once. Of course, going under mountains or to abandonedmines, using radiopure materials and building shieldings is of uttermostimportance to be able to reject the environmental backgrounds.

6.2 The annual modulation signal 69

Figure 6.2: Representation of the Earth’s orbit around the Sun that givesrise to the annual modulation signal of DM.

One of the most powerful ways to discriminate between backgrounds (notall unfortunately, as will be explained below) and DM signals is observing anannual modulation.

As can be seen in figure 6.2, due to the motion of the Earth around the Sun,the velocity of the DM will be different throughout the year. Qualitatively,the maximum will be in June, when 𝑣𝐽𝑢𝑛𝑒WIMP wind ∼ 𝑣𝑠𝑢𝑛 + 𝑣𝑒

2 , and the minimumin December, when 𝑣𝐷𝑒𝑐WIMP wind ∼ 𝑣𝑠𝑢𝑛 − 𝑣𝑒

2 . This is known as the annualmodulation signal of DM, and it is a very useful discriminant of backgrounds.We recommend the reader the interesting review on the annual modulation[181].

The halo integral, eq. 6.10, and thus the rate, eq. 6.11, will have time-independent and time-dependent components

𝜂(𝑣𝑚, 𝑡) = 𝜂(𝑣𝑚) + 𝛿𝜂(𝑣𝑚, 𝑡) , (6.12)

In general, one can expand 𝜂(𝑣𝑚, 𝑡) as a Fourier series

𝜂(𝑣𝑚, 𝑡) =∑𝑛=0

𝐴(𝑛)𝜂 (𝑣𝑚) cos 2𝑛𝜋(𝑡− 𝑡𝑛), (6.13)

where 𝐴(0)𝜂 ≡ 𝜂(𝑣𝑚) is the time averaged rate, 𝐴(1)

𝜂 ≡ 𝐴𝜂(𝑣𝑚) is the amplitudeof the annual modulation signal, 𝐴(2)

𝜂 the amplitude of the semi-annualmodulation, and so on.

In the work included in this thesis [4], we were able to bound 𝐴𝜂 in terms


of 𝜂(𝑣𝑚), by doing an expansion in 𝑣𝑒. The bounds have the form∫ 𝑣2

𝑣1𝑑𝑣𝑚𝐴𝜂(𝑣𝑚) ≤ 𝑣𝑒 𝜂(𝑣1) , (6.14)

and are almost independent of astrophysics.

6.3 The velocity distribution and the 𝜂-𝑣minplot

Astrophysical uncertainties are very relevant to direct detection experiments(see for instance [182–184]).

While stars (baryons) travel together in roughly circular orbits withsmall velocity dispersion, and therefore have roughly the same velocity inthe galactic frame, ∼ 220 km s−1, DM particles travel individually with nocircular dependence in general and therefore have a large velocity dispersion,with a wide velocity distribution.

When presenting results of DM direct detection experiments, it has becomecustomary to use the Standard Halo Model (SHM), i.e., a Maxwellian DMvelocity distribution, in order to show the constraints on the plane scatteringcross section versus DM mass. This assumes an isotropic, isothermal spherefor the DM distribution. In the galactic rest frame (truncated at the galaxyescape velocity 𝑣esc to account for the fact that very high velocity WIMPsescape the Galaxy’s potential and therefore the tail of the distribution isdepleted) it is given by:

𝑓gal(v) ∝ [𝑒− v2𝑣2 − 𝑒− 𝑣2

esc𝑣2 ]Θ(𝑣esc − 𝑣) , (6.15)

where the second term is just to smoothen the transition near the escapevelocity. Typically one uses 𝑣 = 220 km s−1 and 𝑣esc = 550 km s−1. This is anextra source of uncertainty, giving the maximum speed of DM particles inthe detector rest frame:

𝑣maximum ∼ 𝑣sun + 𝑣esc. (6.16)

There is some uncertainty in 𝑣sun, but most of it comes from 𝑣esc, whichrecent measurements from RAVE [186] set to 533+54

−41 km s−1. It is importantto notice that, as we saw, for light WIMPs 𝑣𝑚 ∝ 1/𝑚𝜒 and therefore 𝑣esc canaffect the limits of heavy targets on light WIMPs.

However, the most important astrophysical uncertainties do not comefrom 𝜌𝜒 or 𝑣esc, but from the velocity distribution 𝑓gal(v). This SHM is clearly

6.3 The velocity distribution and the 𝜂-𝑣min plot 71

Figure 6.3: Velocity distribution function from N-body simulations of ViaLactea II (from [185]). The right (left) panels are in the restframe of theEarth on June 2nd, when the Earth’s velocity relative to Galactic DM halois maximum (of the galaxy). The solid red line is the average distribution,the light and dark green shaded regions denote the 68% scatter around themedian and the minimum and maximum values over the samples, and thedotted line represents the best-fitting Maxwell-Boltzmann distribution.

an oversimplification, with N-body simulations indicating a more complicatedstructure of the DM halo, see e.g., fig. 6.3 (from [185]). In fact, from N-body simulations, we know that, in the tail of the DM velocity distribution,halo-substructures such as tidal streams [178, 187–189]- regions with verysmall velocity dispersions, not spatially mixed (they can be parametrized by𝛿3(��− 𝑣𝑠))- or debris flows [190], spatially homogeneous regions with constantspeed (which can be parametrized by 𝛿(𝑣 − 𝑣𝑠)) are expected. The true


distribution is thus quite sensitive to the exact history of the MW, mergers,etc, and significantly depends on the halo properties, for instance with thepresence of a dark disk [191] or extragalactic components [192].

Figure 6.4: 𝑓(𝑣) (top) and 𝜂(𝑣𝑚) (bottom) versus velocity for the Maxwellianand for a stream (taken from [181]).

The dependence of the WIMP signals on the velocity distribution hasbeen studied for instance in [179,193–196]. In the constant rate, the velocitydistribution is integrated over 𝑣, so it is not so much dependent. However,the annual modulation signals can be greatly modified.

We show in figures 6.4, 6.5 and 6.6 (from [181]) the differences both in therate and in the annual modulation expected depending on the very differentforms of 𝑓(𝑣).

Figure 6.4 plots 𝑓(𝑣) (top) and 𝜂(𝑣𝑚) (bottom) versus velocity, for aMaxwellian velocity distribution and for a stream, while figure 6.5 plots therate 𝜂(𝑣𝑚) and the modulation amplitude 𝐴𝜂(𝑣𝑚) versus the recoil energy𝐸𝑛𝑟 for the Maxwellian, for a stream and for a debris flow. One can see infigure 6.5 that there is a sign flip in 𝐴𝜂(𝑣𝑚) for the SHM (and also for debrisflows and for the stream, although for this last case only at the particularvelocity of the stream the modulation is significant) at some particular recoilenergy 𝐸flip

𝑛𝑟 : that is a clear sign of DM, as backgrounds are not expected toshow such behaviour.

6.3 The velocity distribution and the 𝜂-𝑣min plot 73

Figure 6.5: The constant rate 𝜂(𝑣𝑚) (top) and the modulation amplitude𝐴𝜂(𝑣𝑚) (bottom) versus recoil energy 𝐸𝑛𝑟 for the Maxwellian, for a streamand for a debris flow, shown at the times when the rate is minimized andmaximized (taken from [181]).

In figure 6.6, the fractional modulation (modulation over constant rate) isplotted versus time for the Maxwellian, for a stream (reduced by ∼ 1/10) andfor the sum of both contributions. Non-sinusoidal behaviours are expectedwhen one considers a non SHM. From the recoil energy at which the flipoccurs, 𝐸flip

𝑛𝑟 , related to the minimum velocity at which it occurs, that for theSHM is 𝑣flip

𝑚 ∼ 200 km s−1, one can infer the DM mass 𝑚𝜒 via eq. 6.6.Presumably, the true velocity distribution is a mixture of different con-

tributions, i.e., a SHM with some streams and debris here and there, andone should try to be completely independent of it when comparing differentexperiments, which in addition are sensitive to different velocity space regionsof 𝑓(𝑣) (their 𝑣𝑚 ranges can be completely different).

Therefore, showing the experimental data in an astrophysics - independentfashion is of uttermost interest. The halo integral 𝜂(𝑣𝑚, 𝑡), eq. 6.10, is thebasis for the astrophysics-independent comparison of experiments [197,198].It uses an extremely simple and useful observation: one can translate differentenergy ranges 𝐸𝑛𝑟 𝑖 of different experiments 𝑖 into a 𝑣𝑚 range common to all ofthem, and plot their corresponding 𝜂(𝑣𝑚, 𝑡), which encodes the astrophysicsuncertainty on 𝑣esc and 𝑓(��) and has to be the same for all experiments.In [199], this halo-independent method to compare data from DM direct


Figure 6.6: The fractional modulation versus time for the Maxwellian, fora stream (reduced by ∼ 1/10 and for the sum of both contributions (takenfrom [181]).

detection was extended to any type of interaction, like magnetic momentones.

In the work included in this thesis [5], using this fact and applying theastrophysics-independent bounds derived in our first work [4], we were ableto bound the modulation amplitude of one experiment, 𝐴exp 1

𝜂 , in terms of theconstant rate of another experiment, 𝜂(𝑣𝑚)exp 2. In [6] we extended the studyto inelastic scattering.

6.4 Current experimental situation 75

6.4 Current experimental situationFor SD the leading constraints come from SIMPLE [200], PICASSO [201],and COUPP [202] and for SI from CDMS-Ge [203,204], XENON100 [205],and more recently LUX [206], which has put the most stringent bounds on SIinteractions of DM.

Some experiments have found anomalies that may be due to a DM particlewith a low mass (. 10 GeV), like DAMA [207], CoGeNT [208], and CRESST[209]. The situation is also controversial with CDMS Si [210], which has threeevents that can be interpreted as DM. All of these are in strong tension (ifnot excluded) with several of the previous null-results.

The two experiments that see an annual modulation are DAMA [207], at∼ 9𝜎 and CoGeNT [208], at 2.2𝜎, with compatible phases between themselves.However, the amplitude of the modulation in the CoGeNT is much largerthan its rate, pointing if interpreted as DM to a non-Maxwellian halo ornon-standard DM interactions. We analysed the compatibility of modulationsand rates within the same experiment in great detail in the research donein [4].

The DAMA/LIBRA results [207] have a total exposure of 1.17 ton yr over13 years with a modulation claimed to be compatible with DM at the ∼ 9𝜎level. This annual modulation can be explained under standard assumptionsregarding the DM halo (SHM) and WIMP interactions (elastic, SI) with aDM mass of either ∼ 10 scattering primarily on Na nuclei or of 80 GeVscattering primarily on I nuclei, see Figure 6.7, from [211] (and also forinstance [196,212]),2 with this last solution being excluded by LUX by severalorders of magnitude [206]. DAMA claims that no modulating backgroundsatisfies having a sinusoidal one-year period rate with the proper phase andamplitude, and present only in a definite low energy range.

Note also that gravitational focusing of the Sun affects the phase of themodulation [213–216], especially at low velocities (large DM masses), and themodulation caused just by this effect has minima (maxima) when the Earth isbehind (in front of) the Sun, which occurs in March (September). Thereforethe standard modulation phase caused by the Earth’s velocity predicted inthe SHM, June 1st, changes due to gravitational focusing. For DAMA, inthe energy ranges 2 − 4, 2 − 5, 2 − 6 keVee were data is available, it changesby a few days for an 80 GeV DM mass, being predicted to be around May20th [216], which is compatible with the one observed by the collaboration.

2Really there are in both cases more scatterings on iodine due to the 𝐴2 enhancementin the cross section, however for a 10 GeV particle these are below the energy threshold.Lowering the threshold of the DAMA experiment would in principle allow to distinguishbetween both solutions [212].


Many explanations have been proposed in the literature for the DAMAsignal:

∙ Radon abundance, with an increase in the summer and a decrease inthe winter.

∙ Neutron flux, that modulates due to the water and snow present inthe overburden (in winter the water absorbs more neutrons than inthe summer) or the change in density of the upper atmosphere. Forinstance, in [217], neutron data were extrapolated to the DAMA region,and in principle could give such an annual modulation.

∙ Cosmic muons. For instance, in [218], using muon data from MACRO,LVD and Borexino, the muon phase is found to be incompatible withthat of the DAMA at 5.2𝜎.

A possibility is that DAMA might be seeing a combination of these effects,and therefore observing a mixed phase, quite close to the SHM DM one.

Figure 6.7: WIMP discovery limit (thick dashed orange) compared withcurrent limits and regions of interest from [211]. The dominant neutrinocomponents for different WIMP mass regions are labeled. The regions andbounds shown are at 90% C.L.

In fig. 6.7, taken from [211], the current experimental limits are summa-rized, together with the irreducible neutrino background that next generation

6.4 Current experimental situation 77

experiments will be facing. Progress beyond the cross sections of this neutrinobackground will require discriminating it from DM via annual modulation(although it also modulates) or directional detection signals.

Part IV

Final remarks and conclusions

Final remarks and conclusions 81

On radiative neutrino mass modelsWe summarize in this section the results presented in the first part of thethesis, which is devoted to the study of neutrino masses via radiative models.

As has been argued in chapters 2 and 3, there are several mechanismsproposed to give masses to neutrinos. A very appealing option are radiativemodels, which explain why neutrino masses are much smaller than the massesof the rest of the fermions: neutrinos are massless at tree level, with their massbeing generated radiatively at one, two or three loops (more loops typicallyyield too light neutrino masses and/or are generally incompatible with low-energy constraints, like charged LFV constraints). In addition, thanks to theloop suppression and the mandatory presence of a few couplings to violatelepton number (LN), these models have particles at a scale low enough to beproduced at colliders, in particular in our case, at the LHC, and in low-energyexperiments, giving signals in lepton flavour violating (LFV) processes like𝜇 → 𝑒𝛾 or LNV ones, such as 0𝜈𝛽𝛽.

In this thesis we have studied radiative generation of neutrino massesat two-loops both through new fermions (new families) and new scalars(the Zee-Babu model). We also studied the implications of these models toother HEP scenarios, either indirectly, such as Higgs physics, for instance viamodifications of the 𝐻 → 𝛾𝛾 signal, or directly, detecting the new particles.

In the first works, we studied in detail the possibility of the existence of afourth SM sequential generation, addressing in particular the light neutrinoproperties and their nature [1]. The fourth generation neutrino must be muchheavier than the light ones, 𝑚4 & 𝑚Z/2, and a right-handed neutrino is verylikely needed to provide a mass for it. We realised that if light neutrinos areMajorana particles, as LN is then violated, in all types of seesaw models thereis a contribution to the Majorana mass of this right-handed neutrino at twoloops. Therefore, the fourth generation neutrino should also be Majorana,unless some symmetry is imposed on the leptons of the fourth generation.

We analysed in detail the possible implications of having a heavy fourth-generation Majorana neutrino. In particular, it generates light neutrinomasses at two loops and this contribution can easily exceed the cosmologicalbound on the absolute mass scale of the light neutrinos. There can also beextra contributions from the heavy states to 0𝜈𝛽𝛽, which, however, when oneimposes that light neutrino masses do not exceed the cosmological scale, arein most regions of the parameter space smaller than the usual contributionfrom light neutrinos. We also studied possible signals in LFV processes andother rare processes.

However, even if there is a contribution to the light neutrino masses, theirspectrum cannot be explained with just an extra family. Therefore, in a

82 Final remarks and conclusions

different work, we studied the possibility that light neutrinos were masslessat tree level, with their masses being generated at two loops by the action oftwo extra generations [2].

Currently, thanks to the LHC the existence of extra sequential generationsof particles has now been ruled out, at least if the scalar sector comprisesjust the SM Higgs boson, due to the fact that the Higgs boson production isenhanced in the presence of new fermions (and also its decays vary significantlyif a new family exists). Other possibilities arise which may save the fourthgeneration, such as an extended scalar sector, for instance with an extraHiggs doublet that only couples to the fourth family [10]. Limits on fourthgeneration fermions are, however, very close to the perturbative limit in thecase of the quarks & 600 GeV, see for instance [11],

In another work [3], we updated the current status of the Zee-Babu modelin the light of new data, e.g., the mixing angle 𝜃13, the rare decay 𝜇 → 𝑒𝛾and the LHC results. We also analysed the possibility of accommodatingthe deviations in Γ(𝐻 → 𝛾𝛾) hinted by the LHC experiments (an enhancedHiggs diphoton decay rate in the ZB model is compatible with current dataonly in a small part of the parameter space), and the stability of the scalarpotential, as well as restrictions that can be placed on the trilinear couplingfrom charge conservation.

We found that neutrino oscillation data and low energy constraints arestill compatible with masses of the extra charged scalars accessible to LHC.We also studied the decays of the doubly-charged singlets, which could bedetected at LHC-14 and may help to discriminate this model from others.Moreover, if any of the scalars is discovered, the model can be falsified bycombining the information on the singly and doubly charged scalar decaymodes with neutrino data. Conversely, if the neutrino spectrum is found tobe inverted and the CP phase 𝛿 is quite different from ∼ 𝜋, the masses of thecharged scalars will be well outside the LHC reach.

In a broad sense, we have tried to shed some light on different forms togenerate neutrino masses radiatively, focusing on their testability, alwayswith an eye on possible signals at the LHC, and their connection to otherHEP branches. Hopefully, with more experimental input, nature will help usdisentangle between the overwhelming list of possibilities. For the moment,we can say that extra families are in a very bad shape, while the Zee-Babu isstill alive and ready to be discovered at LHC-14!

Final remarks and conclusions 83

On bounds on the DM annual modulation sig-nalWe summarize in this section the results presented in the second part of thethesis, which is devoted to the study of bounds on the annual modulationsignal.

As has been argued in chapter 5, Weakly Interacting Massive Particles(WIMPs) are well-motivated dark matter (DM) candidates which are expectedto talk to the SM, as their relic abundance comes from thermal freeze-out.In this sense, a signal in direct detection experiments could be possible andprobably expected, as discussed in chapter 6, as well as signals in colliders, inindirect detection and in other cosmological observables.

Direct detection experiments are very difficult to perform, as the expectedDM signals are very low and the backgrounds are very large, even whenthe experiments are placed, shielded, in deep-underground laboratories. Infact, as we saw in figure 6.7, we will be reaching the irreducible neutrinobackground in the future.

The annual modulation that should be present in the signal of directdetection experiments, due to the relative motion of the Earth around theSun, could help discriminate a DM signal from the background (some canmodulate, like muons, but the phase does not have to coincide with theexpected one from DM).

An important point to keep in mind when interpreting a DM signal isthat the rates are subject to astrophysical uncertainties, such as the velocitydistribution of the DM particles, the local density or the escape velocity,so it is important to be as independent of the astrophysical parameters aspossible. This can be done by plotting their corresponding inverse meanvelocity 𝜂(𝑣𝑚, 𝑡) in the same range of 𝑣𝑚.

In this thesis, we bounded the annual modulation in terms of the unmodu-lated event rate, which is almost independent from astrophysical uncertainties,by expanding to first order the rate (really 𝜂(𝑣𝑚, 𝑡), see eq. 6.10) in powersof 𝑣𝑒(𝑡)/𝑣, where 𝑣𝑒(𝑡) is the velocity of the Earth around the Sun and 𝑣is the DM velocity in the halo. The method is an important test that anyannually-modulated DM signal has to pass.

We applied these bounds to DAMA and CoGeNT, comparing their modu-lations and rates [4]. DAMA’s modulation is compatible with its own rate,while CoGeNT is excluded for typical DM haloes at & 90% C.L.

In a second work we showed how the unmodulated rate of one experimentcan bound the annual modulation seen in another experiment in the same𝑣𝑚 range. We applied it to DAMA’s modulation versus null-results from

84 Final remarks and conclusions

XENON, CDMS and other experiments, for the case of elastic scattering,with spin-independent (SI), spin-dependent (SD) and isospin violating (IV)interactions [5]. A DM interpretation of DAMA’s modulation was disfavouredby at least one experiment at more than 4𝜎 for a DM mass . 15 GeV, for alltype of interactions.

In particular, for the case of spin-independent interactions, DAMA isexcluded by XENON100 at more than 6𝜎 for a DM mass larger or equal to8 GeV. Of course, the CL of exclusion depends on systematic uncertainties,such as the sodium quenching factor, but even letting it vary, SI interactionsare excluded at more than 5𝜎 for a dark matter mass larger or equal to 10GeV. For the case of SD or IV interactions, consistency can be achieved atabout 3𝜎 for this DM mass.

We have also extended the analysis to inelastic scattering [6], showing howDAMA is incompatible with XENON 100 null-results. This extension was nottrivial, due to the fact that at larger velocities, where inelastic scattering isrelevant, substructure in the DM halo, such as cold streams, may be present,and higher harmonics can become important. We also applied the shapetest to DAMA, which comes from the fact that in inelastic scattering therates of the two different energy branches within an experiment (for instance,DAMA), corresponding to the same 𝑣𝑚, should be equal. In some regionsof the parameter space inelastic scattering for DAMA can be excluded justbased on the energy spectrum of the modulation.

In a broad way, we have derived astrophysics-independent bounds thatannual modulations have to fulfill. Hopefully, this kind of tests will help todiscriminate between true DM signals and backgrounds. It is important toemphasize that it is a necessary condition for the modulation to be of DMorigin, but not sufficient.

After applying our bounds to DAMA and CoGeNT annual modulations’and comparing them to other null-result experiments, they are in very badshape. We expect that in future years more experimental input from directdetection experiments will help to clarify the currently confusing situationamong these different experimental results, that look really incompatible, andlet’s hope that DM is out there ready to be discovered one way or another!

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Part V

Scientific Research

FTUV-14-0214, IFIC/14-13

The Zee-Babu Model revisited in the light of new data

Juan Herrero-Garciaa, Miguel Nebotb, Nuria Riusa and Arcadi Santamariaa

a Departament de Fısica Teorica, Universitat de Valencia and

IFIC, Universitat de Valencia-CSIC

Dr. Moliner 50, E-46100 Burjassot (Valencia), Spain and

b Centro de Fısica Teorica de Partıculas,

Instituto Superior Tecnico – Universidade de Lisboa

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Abstract

We update previous analyses of the Zee-Babu model in the light of new data, e.g., the mixing angle

θ13, the rare decay µ→ eγ and the LHC results. We also analyse the possibility of accommodating

the deviations in Γ(H → γγ) hinted by the LHC experiments, and the stability of the scalar

potential. We find that neutrino oscillation data and low energy constraints are still compatible

with masses of the extra charged scalars accessible to LHC. Moreover, if any of them is discovered,

the model can be falsified by combining the information on the singly and doubly charged scalar

decay modes with neutrino data. Conversely, if the neutrino spectrum is found to be inverted and

the CP phase δ is quite different from π, the masses of the charged scalars will be well outside the

LHC reach.

1

I. INTRODUCTION

The observed pattern of neutrino masses and mixing remains one of the major puzzles

in particle physics. Moreover, massive neutrinos provide irrefutable evidence for physics

beyond the Standard Model (SM) and many theoretical possibilities have been proposed

to account for the lightness of neutrinos (see [1–4] for some reviews). With the running of

the LHC, it is timely to explore neutrino mass models in which the scale of new physics is

close to the TeV. In particular, radiative mechanisms are especially appealing, since small

neutrino masses are generated naturally due to loop factors. On the other hand, new physics

effects can be sizable also in low energy experiments, for instance lepton flavour violating

rare decays of charged leptons, `α → `βγ, providing complementary probes for such models.

In this paper we consider the Zee-Babu model (ZB) of neutrino masses1, which just adds

two (singly and doubly) charged scalar singlets to the SM. Neutrino masses are generated

at two loops and are proportional to the Yukawa couplings of the new scalars and inversely

proportional to the square of their masses. This is phenomenologically quite interesting

because the new scalars cannot be very heavy or have very small Yukawa couplings, otherwise

neutrino masses would be too small. As a consequence, such scalars may be accessible at

the LHC, and in principle they could explain the slight excess over the SM prediction found

by ATLAS in the diphoton Higgs decay channel H → γγ (currently CMS does not see any

excess, see section III for the latest data). They also mediate a variety of lepton flavour

violating (LFV) processes, leading to rates measurable in current experiments.

The phenomenology of the ZB model has been widely analyzed: neutrino oscillation data

was used to constrain the parameter space of the model, LFV charged lepton decay rates

calculated and collider signals discussed [10–12]. Non-standard neutrino interactions in the

ZB model have also been thoroughly studied, in correlation with possible LHC signals and

LFV processes [13]. In [12], some of us performed an exhaustive numerical study of the full

parameter space of the model using Monte Carlo Markov Chain (MCMC) techniques, which

allow to efficiently explore high-dimensional spaces. However, in the last few years there

1 The model was first proposed in [5] and studied carefully in [6]. Similar models with a doubly charged

scalar and masses generated at two loops were discussed in [7] (two-loop neutrino mass models containing

doubly-charged singlets have also been recently discussed in connection with neutrinoless double beta

decay [8, 9]).

2

have been several experimental results which motivate an up-to-date analysis including all

relevant data currently available. Therefore, in this work we update previous analysis in

the light of the recent measurement of the neutrino mixing angle θ13 [14–16], the new MEG

limits on µ → eγ [17], the lower bounds on doubly-charged scalars coming from LHC data

[18, 19], and, of course, the discovery of a 125 GeV Higgs boson by ATLAS and CMS [20, 21].

Moreover, we also study the possibility of accommodating deviations from the SM prediction

for the Higgs diphoton decay channel, and the effects of the new couplings of the model in

the stability of the scalar potential. A possible enhancement of the Higgs diphoton decay

rate in the ZB model together with the vacuum stability of the scalar potential has been

studied in [22], however a consistent updated analysis including all constraints is lacking.

The outline of the paper is the following. In section II we briefly review the main features

of the ZB model, discussing perturbativity and naturality estimates for the allowed ranges

of the free parameters of the model. We summarize present constraints from recent neutrino

oscillation data, low energy lepton-flavour violating processes, universality and stability of

the scalar potential. We also review the collider phenomenology of the ZB model, discussing

current limits from LHC, and briefly comment on the prospects for non-standard neutrino

interactions. In section III we analyze in detail the contributions of the ZB charged scalars to

both, Γ(H → γγ) and Γ(H → Zγ). After some analytic estimates in section IV, we present

the results of our MCMC numerical analysis in section V and we conclude in section VI.

Renormalization group equations for the ZB model and relevant loop functions are collected

in the appendices.

II. THE ZEE-BABU MODEL

We follow the notation of [12]. As mentioned above, the Zee-Babu model only contains,

in addition to the SM, two charged singlet scalar fields

h±, k±± , (1)

with weak hypercharges ±1 and ±2 respectively (we use the convention Q = T3 + Y ).

The scalar potential is given by

V = m′2HH†H +m′2h |h|2 +m′2k |k|2 + λH(H†H)2 + λh|h|4 + λk|k|4

+ λhk|h|2|k|2 + λhH |h|2H†H + λkH |k|2H†H +(µh2k++ + h.c.

), (2)

3

being H the SU(2) doublet Higgs boson, while the leptons have Yukawa couplings to both

H and the new charged scalars:

LY = LL Y eH + LLf`h+ + ecg e k++ + h.c. , (3)

where LL and e are the SM SU(2) lepton doublets and singlets, respectively, and LL ≡

iτ2LcL = iτ2CLL

T, with τ2 Pauli’s second matrix. Due to Fermi statistics, fab is an antisym-

metric matrix in flavour space while gab is symmetric.

Notice that we can assign lepton number −2 to both scalars, h+ and k++, in such a way

that total lepton number L (or B−L) is conserved in the complete Lagrangian, except for the

trilinear coupling µ of the scalar potential; thus, lepton number is explicitly broken by the µ-

coupling. It is important to remark that lepton number violation requires the simultaneous

presence of the four couplings Y , f , g and µ, because if any of them vanishes one can always

assign quantum numbers in such a way that there is a global U(1) symmetry. This means

that neutrino masses will require the simultaneous presence of the four couplings.

Regarding the physical free parameters in the ZB model, our convention is the following:

without loss of generality, we choose the 3 × 3 charged lepton Yukawa matrix Y to be

diagonal with real and positive elements. We also use fermion field rephasings to remove

three phases from the elements of the matrix g and charged scalar rephasings to set µ real

and positive, and to remove one phase from f . In summary we have 12 moduli (3 from

Y , 3 from f and 6 from gab), 5 phases (3 from g and 2 from f) and the real and positive

parameter µ, plus the rest of real parameters in the scalar potential. As discussed in [12],

this choice is compatible with the standard parametrization of neutrino masses and mixings.

After electroweak symmetry breaking, the masses of charged leptons are ma = Yaav, with

v ≡ 〈H0〉 = 174 GeV, the VEV of the standard Higgs doublet, while the physical charged

scalar masses are given by

m2h = m′2h + λhHv

2 , m2k = m′2k + λkHv

2 . (4)

In principle, the scale of the new mass parameters of the ZB model (mh,mk and µ)

is arbitrary. However from the experimental point of view it is interesting to consider

new scalars light enough to be produced in the second run of the LHC. Also theoretical

arguments suggest that the scalar masses should be relatively light (few TeV), to avoid

unnaturally large one-loop corrections to the Higgs mass which would introduce a hierarchy

4

problem. Therefore, in this paper we will focus on the masses of the new scalars, mh,mk,

below 2 TeV.

The Yukawa couplings of the new scalars of the model enter in the neutrino mass for-

mula and in several LFV processes, and are strongly bounded for the scalar masses we are

considering except in a few corners of the parameter space where we require that the theory

remains perturbative. Since one-loop corrections to Yukawa couplings are order

δf ∼ f 3

(4π)2, δg ∼ g3

(4π)2. (5)

one expects from perturbativity f, g � 4π, although, as we will see, for the scalar masses

considered here, phenomenological constraints are always stronger.

The couplings of the charged scalars in the scalar potential, apart from the stability

constraints described in section II E, are essentially free. However, for the theory to make

sense as a perturbative theory we also impose the limit2 λh,k,kH,hH,hk < 4π.

The trilinear coupling among charged scalars µ, on the other hand, is different, for it has

dimensions of mass and it is insensitive to high energy perturbative unitarity constraints.

However, it induces radiative corrections to the masses of the charged scalars of order

δm2k, δm

2h ∼

µ2

(4π)2. (6)

Requiring that the corrections in absolute value are much smaller than the masses we can

derive a naive upper bound for this parameter, µ� 4πmin(mh,mk), but it is difficult to fix

an exact value of µ for which the contributions to the scalar masses are unacceptably large,

leading to a highly fine-tuned scenario.

A large value of µ, as compared with scalar masses, is also disfavoured because it could

lead to a deeper minimum of the scalar potential for non-vanishing values of the charged

fields, therefore breaking charge conservation. This phenomenon has also been studied in the

context of supersymmetric theories (see for instance [25–27]). As an example, by looking

at the particular direction |H| = |h| = |k| = r, and requiring that the charge breaking

minimum is not a global minimum, V (r 6= 0) > 0, one obtains

µ2 < (λH + λh + λk + λhH + λkH + λhk)(m′2H +m′2h +m′2k

). (7)

2 Notice that there could be order one differences in the perturbativity constraints on the different couplings

λi from perturbative unitarity of the matrix elements [23, 24]. We can neglect them for the purpose of

this work, keeping in mind that they could be relevant when perturbativity is “pushed” to the limit (as

needed to explain H → γγ enhancement, see sec. III).

5

Figure 1: Diagram contributing to the neutrino Majorana mass at two loops.

Assuming no cancellations between the λ’s or mass terms, neglecting λH and m′2H , and using

the perturbative limit for the rest of the couplings λi <∼ 4π one finds a very conservative

bound on µ

µ <∼√

20πmax(mk,mh) ∼ 8 max(mk,mh) (8)

Tighter limits can be obtained by looking at all directions in the potential and/or allowing

for cancellations.

Given that the neutrino masses depend linearly on the parameter µ, as we will see in

the next section, the ability of the model to accommodate all present data is quite sensitive

to the upper limit allowed for µ. Thus we choose to implement such limit in terms of a

parameter κ,

µ < κmin(mh,mk) , (9)

and discuss our results for different values of κ = 1, 5, 4π. Notice that we are using the

naturality upper bound (expressed in terms of min(mh,mk)), which in general is much more

restrictive than the upper bound obtained by requiring that the minimum of the potential

does not break charge conservation (expressed in terms of max(mh,mk)).

A. Neutrino masses.

The lowest order contribution to neutrino masses involving the four relevant couplings

appears at two loops [5, 6] and its Feynman diagram is depicted in fig. 1.

The calculation of this diagram gives the following mass matrix for the neutrinos (defined

6

as an effective term in the Lagrangian Lν ≡ −12νcLMννL + h.c.)

(Mν)ij = 16µfiamag∗abIabmbfjb , (10)

where Iab is the two-loop integral, which can be calculated analytically [28]. However, since

mc,md are the masses of the charged leptons, necessarily much lighter than the charged

scalars, we can neglect them and obtain a much simpler form

Icd ' I =1

(16π2)2

1

M2

π2

3I(r) , M ≡ max(mh,mk) , (11)

where I(r) is a function of the ratio of the masses of the scalars r ≡ m2k/m

2h,

I(r) =

1 + 3π2 (log2 r − 1) for r � 1

1 for r → 0, (12)

which is close to one for a wide range of scalar masses. Within this approximation the

neutrino mass matrix can be directly written in terms of the Yukawa coupling matrices, f ,

g, and Y

Mν =v2µ

48π2M2I f Y g†Y TfT . (13)

A very important point is that since f is a 3 × 3 antisymmetric matrix, det f = 0 (for

3 generations), and therefore detMν = 0. Thus, at least one of the neutrinos is exactly

massless at this order.

The neutrino Majorana mass matrix Mν can be written as

Mν = UDνUT , (14)

where Dν is a diagonal matrix with real positive eigenvalues, and U is the Pontecorvo-Maki-

Nakagawa-Sakata (PMNS) leptonic mixing matrix. We are left with only two possibilities

for the neutrino masses, mi:

• Normal hierarchy (NH): the solar squared mass difference is ∆S = m22, the atmospheric

mass splitting ∆A = m23 and m1 = 0, with m3 � m2 .

• Inverted hierarchy (IH): ∆S = m22 −m2

1, ∆A = m21 and m3 = 0, with m1 ≈ m2.

The standard parametrization for the PMNS matrix is

U =

c13c12 c13s12 s13e

−iδ

−c23s12 − s23s13c12eiδ c23c12 − s23s13s12e

iδ s23c13

s23s12 − c23s13c12eiδ −s23c12 − c23s13s12e

iδ c23c13

1

eiφ/2

1

, (15)

7

Process Experiment (90% CL) Bound (90% CL)

µ− → e+e−e− BR< 1.0× 10−12 |geµg∗ee| < 2.3× 10−5(mkTeV

)2τ− → e+e−e− BR< 2.7× 10−8 |geτg∗ee| < 0.009

(mkTeV

)2τ− → e+e−µ− BR< 1.8× 10−8 |geτg∗eµ| < 0.005

(mkTeV

)2τ− → e+µ−µ− BR< 1.7× 10−8 |geτg∗µµ| < 0.007

(mkTeV

)2τ− → µ+e−e− BR< 1.5× 10−8 |gµτg∗ee| < 0.007

(mkTeV

)2τ− → µ+e−µ− BR< 2.7× 10−8 |gµτg∗eµ| < 0.007

(mkTeV

)2τ− → µ+µ−µ− BR< 2.1× 10−8 |gµτg∗µµ| < 0.008

(mkTeV

)2µ+e− → µ−e+ GMM < 0.003GF |geeg∗µµ| < 0.2

(mkTeV

)2Table I: Constraints from tree-level lepton flavour violating decays [3].

where cij ≡ cos θij, sij ≡ sin θij and since one of the neutrinos is massless, there is only one

physical Majorana phase, φ, in addition to the Dirac phase δ.

B. Low energy constraints.

In order to provide neutrino masses compatible with experiment, the Yukawa couplings

of the charged scalars cannot be too small and their masses cannot be too large. This

immediately gives rise to a series of flavour lepton number violating processes, as for instance

µ− → e−γ or µ− → e+e−e−, with rates which can be, in some cases, at the verge of the

present experimental limits. Therefore, we can use these processes to obtain information

about the parameters of the model and hopefully to confirm or to exclude the model in a

near future by exploiting the synergies with direct searches for the new scalars at LHC.

In this section we follow the notation of [12], where all the relevant formulae can be

found, and update the new bounds. We collect the relevant tree-level lepton flavour violating

constraints, from `−a → `+b `−c `−d decays and µ+e− ↔ µ−e+ transitions, in table I.

Universality constraints are summarized in table II where we have combined the mea-

surements presented in [29] for the different couplings. There seems to be a 2σ discrepancy

in Gexpτ /Gexp

e , which we interpret as a bound. If confirmed and interpreted within the ZB

model, one obtains that |fµτ |2 − |feµ|2 = 0.05 (mh/TeV)2. As we will see in section IV, for

NH spectrum feµ ∼ fµτ/2, therefore one needs mh ∼ 4 fµτTeV, which is easily achieved. For

8

SM Test Experiment Bound (90%CL)

lept./hadr. univ.∑

q=d,s,b |Vexpuq |2 = 0.9999± 0.0006 |feµ|2 < 0.007

(mhTeV

)2µ/e universality

Gexpµ

Gexpe= 1.0010± 0.0009 ||fµτ |2 − |feτ |2| < 0.024

(mhTeV

)2τ/µ universality Gexpτ

Gexpµ= 0.9998± 0.0013 ||feτ |2 − |feµ|2| < 0.035

(mhTeV

)2τ/e universality Gexpτ

Gexpe= 1.0034± 0.0015 ||fµτ |2 − |feµ|2| < 0.04

(mhTeV

)2Table II: Constraints from universality of charged currents obtained combining the experimental

results compiled in table 2 of [29].

Experiment Bound (90%CL)

δae = (12± 10)× 10−12 r(|feµ|2 + |feτ |2

)+ 4

(|gee|2 + |geµ|2 + |geτ |2

)< 5.5× 103 (mk/TeV)2

δaµ = (21± 10)× 10−10 r(|feµ|2 + |fµτ |2

)+ 4

(|geµ|2 + |gµµ|2 + |gµτ |2

)< 7.9 (mk/TeV)2

BR(µ→ eγ) < 5.7× 10−13 r2|f∗eτfµτ |2 + 16|g∗eegeµ + g∗eµgµµ + g∗eτgµτ |2 < 1.6× 10−6 (mk/TeV)4

BR(τ → eγ) < 3.3× 10−8 r2|f∗eµfµτ |2 + 16|g∗eegeτ + g∗eµgµτ + g∗eτgττ |2 < 0.52 (mk/TeV)4

BR(τ → µγ) < 4.4× 10−8 r2|f∗eµfeτ |2 + 16|g∗eµgeτ + g∗µµgµτ + g∗µτgττ |2 < 0.7 (mk/TeV)4

Table III: Constraints from loop-level lepton flavour violating interactions and anomalous magnetic

moments [3, 17].

IH spectrum, however, fµτ ∼ 0.2feµ (fµτ ∼ (0.15 − 0.3) feµ if we vary the angles in their

3σ range), and therefore, if this measurement is confirmed, the IH scheme in the ZB model

would be disfavoured.

Finally, one-loop level lepton flavour violating constraints coming from `−a → `−b γ decays3

and anomalous magnetic moments of electron and muon are collected in table III, including

the recent limit on BR(µ→ eγ) from the MEG Collaboration [17].

Given that lepton number is not conserved, another interesting low energy process that

could arise in the ZB model is neutrinoless double beta decay (0ν2β). However, since the

singly and doubly charged scalars do not couple to hadrons and are singlet under the weak

SU(2) (therefore, do not couple to W gauge bosons), the 0ν2β rate is dominated by the

Majorana neutrino exchange [34] and it is proportional to the |(Mν)ee|2 matrix element. In

3 As was shown in [30], doubly charged scalars can give logarithmic enhanced contributions to muon-

electron conversion in nuclei. Moreover, planned experiments will improve current limits by four orders

of magnitude [31–33]; however, at present, limits are still not competitive with µ→ eγ.

9

the NH case,

(MNHν )ee =

√∆Sc

213s

212e

iφ +√

∆As213 . (16)

Using neutrino oscillation data, one obtains 0.001 <∼ eV|(MNHν )ee| <∼ 0.004 eV and there-

fore it is outside the reach of present and near future 0ν2β decay experiments.

In the IH case,

(MIHν )ee =

√∆A + ∆Sc

213s

212e

iφ +√

∆Ac213c

212 . (17)

Then, 0.01 eV <∼ |(MNHν )ee| <∼ 0.05 eV and, therefore, it is observable in planned 0ν2β decay

experiments.

C. Non-standard interactions.

The heavy scalars of the ZB model induce non-standard lepton interactions at tree level,

which have been thoroughly analyzed in [13]. In particular, by integrating out the singly

charged scalar h+, the following dimension-6 operators are generated:

LNSId=6 = 2√

2GF ερσαβ(ναγ

µPLνβ)(`ργµPL`σ), (18)

where ` refer to the charged leptons and the standard NSI parameters ερσαβ are given by

ερσαβ =fσβf

∗ρα√

2GFm2h

. (19)

Regarding neutrino propagation in matter, the relevant NSI parameters are εmαβ = εeeαβ. Since

the couplings fσβ are antisymmetric, in the ZB model only εmµτ , εmµµ and εmττ are non zero.

NSI can also affect the neutrino production in a neutrino factory, via the processes µ →

eνβνα. Source effects in the νµ → ντ and νe → ντ channels are produced by the NSI

parameters

εsµτ = εeµτe =fµef

∗eτ√

2GFm2h

, (20)

εseτ = εeµµτ =fµτf

∗eµ√

2GFm2h

, (21)

respectively. Notice that εmµτ = −εs∗µτ , since both NSI parameters are related to the couplings

feµ and feτ .

As we discuss in section V, the ratios of Yukawa couplings feµ/fµτ and feτ/fµτ are entirely

determined by the neutrino mixing angles and Dirac phase of the PMNS matrix U – see

10

eqs. (37) and (38) –, so the impact of the improved bounds on BR(µ → eγ) can be easily

estimated: given that the limit is now ∼ 0.05 times smaller than in the study of [13], and

the contribution of the singly charged scalar h+ to BR(µ → eγ) depends on |f ∗eτfµτ |2, the

current constraints on |fαβ| are roughly a factor 2 tighter than before. Therefore, since the

strength of the NSI depends on ερσαβ ∝ fσβf∗ρα, generically we expect that the allowed size

of the NSI is reduced by a factor ∼ 1/4. According to [13]4, this implies that in the most

favorable case of IH neutrino mass spectrum, εseτ and εsµτ are in the range 3× (10−5− 10−4),

which is in a range difficult to probe, but it might be in a future neutrino factory with a ντ

near detector [35].

D. Bounds on the masses of the charged scalars.

Regarding limits on singly-charged bosons decaying to leptons, the best limit still comes

from LEP II, mh > 100 GeV.

ATLAS and CMS have placed limits on doubly-charged boson masses from searches of

dilepton final states, using data samples corresponding to√s = 7 TeV with an integrated

luminosity of 4.7 fb−1 and 4.9 fb−1, respectively [18, 19]. The authors of [36] show that,

with current data at 8 TeV and 20 fb−1, all the bounds are expected to become about

∼ 100 GeV more stringent if no significant signal is seen. Further tests on the nature of the

doubly charged scalar (i.e., singlet or triplet of SU(2)L) can be obtained by analysing tau

lepton decay distributions which are sensitive to the chiral structure of the couplings [37].

The main production mechanisms of doubly-charged bosons at hadron colliders are pair

production via an s-channel exchange of a photon or a Z-boson, and associated production

with a charged boson via the exchange of a W-boson (see [38, 39] for a general analysis

of the production and detection at LHC of doubly charged scalars belonging to different

electroweak representations). In the Zee-Babu model, the associated production is absent,

because the new scalars are SU(2)L singlets.

The ATLAS analysis [18] focuses on the ee, µµ, eµ channels and assumes that the rest of

4 Notice that although the analysis of [13] has been done for κ = 1, the impact on NSI of the new bounds

from BR(µ→ eγ) (and in general from any LFV decay `α → `βγ) is independent of the value of κ chosen,

because they constraint directly |f∗ασfσβ |/m2h, which is the same combination that appears in the NSI

parameters, eq. (19). The only effect of increasing κ may be that a given point (fασ, fσβ ,mh) is able to

fit neutrino masses with smaller gab and therefore possibly lighter mk.

11

the channels can make up to 90% of the total decays. Then, the limits for the Zee-Babu

model are, at the 95% C.L., 322, 306, 310 GeV (151, 176, 151 GeV) for branching ratios of

100% (10%) to the ee, µµ, eµ channels. Notice that in [18] the limits on doubly-charged

bosons coupling to left-handed leptons are applied, in addition to the seesaw type II case, to

the Zee-Babu model. However, this is not so, as the doubly-charged singlets in the Zee-Babu

model are SU(2)L singlets and thus couple only to right-handed leptons, at variance with

the seesaw type II models, where the doubly-charged bosons are SU(2)L triplets and do

couple only to left-handed leptons. Therefore, in the Zee-Babu case they have a reduced

production cross section, due to their different couplings to the Z-boson, around 2.5 times

smaller than for the case of the triplet [40], and less stringent limits apply: for the Zee-Babu

model one should look at the second part of table I of [18], the one for H±±R ≡ k±±.

The CMS Collaboration has searched for doubly-charged bosons which are SU(2)L

triplets, both assuming that they decay to the different dilepton final states `` (` = e, µ, τ)

100% of the times, i.e., BR(k++ → ``) = 1, and also considering several benchmark points

with different branching ratios.

The CMS 95% C.L. limits for pair production of SU(2)L singlets, which is the one relevant

for the Zee-Babu Model, are around 60− 80 GeV less stringent [39, 40]:

• ee, µµ, eµ : 310 GeV,

• eτ, µτ : 220 GeV,

• ττ : 100 GeV.

Note that whenever the branching ratio to ττ is less than 30% (see table I and VI of

[19]), the bounds are ∼ 280 GeV, provided that there is a significant fraction of decays into

light leptons (ee, µµ, eµ).

In the Zee-Babu model the decay width of k±± into same sign leptons is given by

Γ(k → `a`b) =|gab|2

4π(1 + δab)mk . (22)

Since the gab couplings are free parameters, the BRs of the different decay modes are a

priori unknown, so we can not apply directly these bounds. As we will see in the numerical

analysis, section V, once neutrino oscillation data and low energy constraints are taken into

12

account, the branching ratio to ττ is very small in the Zee-Babu model, less than about 1%.

Then, a conservative limit is mk > 220 GeV.

Moreover, in the ZB model for mk > 2mh > 200 GeV, it can happen that the doubly

charged scalar decays predominantly into hh, which can easily escape detection. This way

the constraints from dilepton searches could be evaded. The relevant decay width is given

by

Γ(k → hh) =1

8π

[µ

mk

]2

mk

√1− 4m2

h

m2k

. (23)

Then, even for gab ∼ 1, for mh = 100 GeV and mk = 200 GeV, we have that Γ(k→hh)Γ(k→``) ≥ 1

for µ ≥ mk, which is still natural as long it is not very large. Thus, we take mk ≥ 200 GeV

in the numerical analysis.

E. Stability of the potential.

In this section we consider further constraints on the ZB model parameter space com-

ing from vacuum stability conditions. The Hamiltonian in quantum mechanics has to be

bounded from below, this requires that the quartic part of the scalar potential in eq. (2)

should be positive for all values of the fields and for all scales. Then, if two of the fields H, k

or h vanish one immediately finds5:

λH > 0, λh > 0, λk > 0 . (24)

Moreover the positivity of the potential whenever one of the scalar fields H, h, k is zero

implies

α, β, γ > −1 , (25)

where we have defined

α = λhH/(2√λHλh) , β = λkH/(2

√λHλk) , γ = λhk/(2

√λhλk) . (26)

Eq. (25) constrains only negative mixed couplings, λxH , λhk (x = h, k), since for positive ones

the potential is definite positive and only the perturbativity limit, λxH , λhk <∼ 4π applies.

5 We do not consider the possibility of zero couplings, which can only appear at very specific scales.

13

Finally, if at least two of the mixed couplings are negative, there is an extra constraint,

which can be written as:

1− α2 − β2 − γ2 + 2αβγ > 0 ∨ α + β + γ > −1 . (27)

We have checked that the above conditions, eqs. (24, 25, 27), are equivalent to the ones

derived in [41] for the Zee model, but they differ from the ones used in [22] for the ZB

model, which seem not to be symmetric under the exchange of α, β, γ, as they should. Our

constraints also agree with the results obtained by using copositive criteria (see for instance

[42]).

The discovery of the Higgs boson with mass mH ∼ 125 GeV at the LHC has raised

the interest on the vacuum stability of the SM potential: for the current central values of

the strong coupling constant and the Higgs and top quark masses, the Higgs self-coupling

λH would turn negative at a scale Λ ∼ 1010 − 1013 GeV [43], indicating the existence of

new physics beyond the SM below that scale. In fact, by using state of the art radiative

corrections, the authors of [43] find that absolute stability of the SM Higgs potential up to

the Planck scale is excluded at 98% C.L. for mH < 126 GeV.

The one-loop renormalization group equations (RGEs) in the ZB model are written in

Appendix A. For a given set of parameters defined at the electroweak scale, and satisfying

the stability conditions discussed above, we calculate the running couplings numerically by

using one-loop RGEs. From eqs. (A1), we see that the new scalar couplings λhH , λkH always

contribute positively to the running of the Higgs quartic coupling λH , compensating for the

large and negative contribution of the top quark Yukawa coupling. Therefore, the vacuum

stability problem can be alleviated in the ZB model with λH remaining positive up to the

Planck scale for the present central values of mt and mH if λxH are not extremely small

(λxH ∼ ±0.2 are enough to stabilize λH maintaining stability/perturbativity of all couplings

up to the Planck scale (see fig. 2)).

On the other hand, as we discuss in section III, the slight excess in the Higgs diphoton

decay channel found at LHC can be accommodated in the ZB model with relatively light

singlet scalars and large, negative, mixed couplings λhH , λkH . However for such values of the

scalar couplings at the electroweak scale, the RGEs lead to vacuum instability (2√λHλx +

λxH < 0, x = h, k) and/or non-perturbativity (λx > 4π) well below the Planck scale. This

can be seen in fig. 2 where we have performed a complete scan of the quartic couplings

14

Figure 2: Allowed regions in λkH vs λk (left) and λhk vs λh (right), taken at the mZ scale, if

perturbativity/stability is required to be valid up to 103, 106, 109, 1012, 1015, 1018 GeV (from light

to dark colours). The rest of the parameters entering the RGE are taken at their measured value

or varied in the range allowed by the perturbativity/stability requirement up to the given scale.

of the scalar potential, run all of them from mZ up to a given scale (µ = 103n GeV with

n = 1, 2, · · · , 6), and check that stability (as explained before) and perturbativity (λi < 4π)

are satisfied at all scales below µ. On the left we represent the region allowed in the λkH–

λk plane, with λ’s taken at the mZ scale, when stability/perturbativity is imposed up to

the different scales µ. Lighter regions correspond to small scales and obviously include the

regions of larger scales. A similar plot is obtained for λhH vs λh. On the right we present

the equivalent results for the couplings λhk vs λh.

III. H → γγ AND H → Zγ

It remains an open question whether the 125 GeV Higgs boson discovered by ATLAS [20]

and CMS [21] is the SM one or has some extra features coming from new physics. While

all the present measurements of the Higgs properties are consistent with the SM values, the

uncertainties are still large, so there is plenty of room for non-standard signals to show up

in the upcoming 13-14 TeV run data. Moreover, the present experimental situation of the

H → γγ decay channel is far from clear: although the last reported analysis of the CMS

15

and ATLAS Collaborations on the diphoton signal strength are barely consistent with each

other within 2σ, ATLAS still observes a ∼ 2σ excess over the SM prediction [44], while the

CMS measurement has become consistent with the SM at 1σ [45]:

ATLAS : Rγγ = 1.55+0.33−0.28 ,

CMS : Rγγ = 0.78+0.28−0.26 , MVA analysis (28)

CMS : Rγγ = 1.11+0.32−0.31 . cut based analysis

It is thus worthwhile to explore whether an eventually confirmed deviation from the SM

prediction in the H → γγ channel can be accommodated within the ZB model.

In the SM the H → γγ channel is dominated by the W boson loop contribution, which

interferes destructively with the top quark one. Since the Higgs coupling to photons is

induced at the loop-level, extra charged fermions or scalars with significant couplings to

the Higgs can change drastically the H → γγ channel with respect to the Standard Model

expectations, either enhancing it or reducing it [46]. Moreover, in the absence of direct

signatures of new particles at LHC, the enhanced Higgs diphoton decay rate might provide

an indirect hint of physics beyond the SM.

The value of the H → γγ decay width in the ZB model with respect to the SM one is

given by [46–48]:

Rγγ =Γ(H → γγ)ZBΓ(H → γγ)SM

= |1 + δR(mh, λhH) + 4 δR(mk, λkH)|2 , (29)

where we have defined δR(mx, λxH) for the scalar x with mass mx and coupling to the Higgs

λxH as:

δR(mx, λxH) ≡ λxH v2

2m2x

A0(τx)

A1(τW ) + 43A1/2(τt)

, (30)

with τi ≡ 4m2i

m2H

and the loop functions Ai(x) (i = 0, 1/2, 1) are defined in Appendix B. Notice

that the dominant W contribution is A1(τW ) = −8.32 for a Higgs mass of 125 GeV, while

A0(τh,k) > 0, therefore in order to obtain a constructive interference we need to consider

negative couplings λhH , λkH .

As discussed in sec. II E, stability of the potential imposes that 2√λHλx + λxH > 0, for

x = h, k. Since MH ∼ 125 GeV fixes the value of the Higgs self-coupling to λH ∼ 0.13, it is

immediately apparent that large and negative λxH couplings are going to be in conflict with

stability of the potential, unless we push λx close to the naive perturbative limit (λx < 4π),

16

Figure 3: Rγγ in the presence of a doubly charged particle. Both an enhancement (as seen by

ATLAS [44]) or a suppression (as seen by CMS [45]), can be accommodated. For the same masses

and couplings, the singly-charged produces a smaller enhancement/suppression than the doubly-

charged, due to its smaller charge.

for which −3 <∼ λhH , λkH . Notice that this fact is not a special feature of the ZB model, but

a generic problem of any scenario in which the enhancement of the Higgs diphoton decay

rate is due to a virtual charged scalar.

We can consider three different cases:

• If mh � mk,

Rhγγ ≈ |1 + δR(mh, λhH)|2 ; (31)

• If mk � mh,

Rkγγ ≈ |1 + 4δR(mk, λkH)|2 ; (32)

• If mh ≈ mk ≡ mS, with

RSγγ ≈ |1 + δR(mS, λhH) + 4 δR(mS, λkH)|2 . (33)

For the same masses and couplings of both singlets, the doubly charged produces a larger

enhancement/suppression than the singly-charged, due to its greater charge.

The largest enhancement can happen when both charged scalars are about the same mass

and these masses are low enough. We show in fig. 3 the prediction of the ratio Rγγ when

the doubly charged scalar k dominates, for different values of the coupling with the Higgs,

17

Figure 4: Contour of Rγγ = 1.55 (left) [44] and Rγγ = 0.78 (right) [45] in the presence of a singly

charged and doubly charged particle with the same couplings.

λkH . Both an enhancement (as seen by ATLAS [44]) or a suppression (as seen by CMS [45]),

can be accommodated. In fact, deviations from the SM value are expected, i.e., Rγγ 6= 1,

in particular for below the TeV scale singlets and sizeable scalar couplings. Of course, even

for light singlets it is possible that Rγγ ≈ 1, either because the relevant scalar couplings are

tiny or due to a cancellation between the contributions of the singly charged and the doubly

charged scalars.

In principle, the enhancement Rγγ induced by a singly charged scalar h of similar mass

and coupling to the Higgs λhH ∼ λkH is smaller; however since the lower limit on mh from

LEP II direct searches is weaker mh > 100 GeV, as discussed in the previous section, and

the largest contribution occurs for lower masses, the resulting values of Rγγ for the allowed

range of mh are comparable to the doubly charged case.

We show in fig. 4 the contours of Rγγ = 1.55 (0.78), motivated by the experimental

results of ATLAS and CMS [44, 45], in the plane of the singly and doubly charged masses,

for various negative (positive) couplings. In summary, to obtain Rγγ ∼ 1.5 we need mh<∼

200 GeV and/or mk<∼ 300 GeV. As it will be shown in the numerical analysis section, these

scalar masses are in tension with describing neutrino oscillation data and being compatible

with current low-energy bounds in the ZB model if naturality is required at the level of

κ = 1, especially for the NH spectrum. Moreover, the large negative values of the couplings

18

Figure 5: RγZ in the presence of a doubly charged particle. As can be seen, H → Zγ is anticorre-

lated with respect to H → γγ.

λxH ∼ −2 required to obtain such enhancement also induce vacuum instability of the ZB

scalar potential, unless the corresponding coupling λx is close to the perturbative limit,

λx ∼ 8.

There is a correlation between H → γγ and H → Zγ [46, 49, 50]. The ratio of the

H → Zγ decay rate in the ZB model with respect to the SM one is:

RZγ =Γ(H → Zγ)ZBΓ(H → Zγ)SM

=

∣∣∣∣∣1− gZhhλhH v2

m2h

A0(τh, λh)

AZγSM− gZkk

2λkH v2

m2k

A0(τk, λk)

AZγSM,

∣∣∣∣∣2

(34)

where AZγSM is the SM H → Zγ decay amplitude,

AZγSM = cot θWA1(τW , λW ) + 6QtT t3 − 2Qts

2W

sW cWA1/2(τt, λt) , (35)

with λi ≡ 4m2i

m2Z

, and the Z boson couplings to the new charged scalars are gZxx = −Qx cot θW ,

x = h, k. The loop functions Ai(x, y) (i = 0, 1/2, 1) can be found in Appendix B.

In fact, to have an enhancement in the H → γγ channel, we need negative couplings of

the singlets with the Higgs, which in turn implies that the H → Zγ channel is reduced with

respect to SM prediction, as can be seen in fig. 5.

IV. ANALYTICAL ESTIMATES

In this section we give some order of magnitude estimates of the free parameters in

the ZB model, which complement and help to understand our full numerical analysis. In

19

particular, we want to estimate to which extent light charged scalar masses, for instance

like those required to fit an enhanced Higgs diphoton decay rate or to have a chance of

being discovered at the LHC, are consistent with neutrino oscillation data and low-energy

constraints.

As discussed in sec. II, with respect to the SM the ZB model has 17 extra parameters

relevant for neutrino masses (9 moduli and 5 phases from the Yukawa couplings f, g, and 3

mass parameters from the charged scalar sector, mh,mk and µ), plus 5 quartic couplings in

the scalar potential. However, some of the free parameters can be traded by the measured

neutrino masses and mixings, ensuring in this way that the experimental data is reproduced

and reducing the number of free variables as follows.

Since det f = 0, there is an eigenvector a = (fµτ ,−feτ , feµ) which corresponds to the zero

eigenvalue, fa = 0 [10]. Then, by exploiting the fact that a is also an eigenvector of Mν ,

we have

DνUTa = 0, (36)

which leads to three equations, one of which is trivially satisfied because one element of Dν

is zero. The other two equations allow to write the ratios of Yukawa couplings fij in terms

of the neutrino mixing angles and Dirac phase as follows:

feτfµτ

= tan θ12cos θ23

cos θ13

+ tan θ13 sin θ23e−iδ ,

feµfµτ

= tan θ12sin θ23

cos θ13

− tan θ13 cos θ23e−iδ , (37)

in the NH case, and

feτfµτ

= − sin θ23

tan θ13

e−iδ ,

feµfµτ

=cos θ23

tan θ13

e−iδ , (38)

for IH spectrum. Therefore, we choose fµτ as a free, real, parameter and obtain (complex) feµ

and feτ from the above equations. Notice that the measured values, s212 ∼ 0.3, s2

23 ∼ 0.4 and

s213 ∼ 0.02 imply that, for NH, the first term on the right-hand side of eqs. (37) dominates

and leads to feµ ∼ fµτ/2 ∼ feτ . Conversely, for IH it is clear that feτ/feµ = − tan θ23 ∼ −1

and |feµ/fµτ | ∼ |feτ/fµτ | ∼ 4. Of course, to explain such fine-tuned relations of Yukawa

couplings a complete theory of flavour would be needed, which is beyond the scope of this

work.

20

Regarding the Yukawa couplings g, we keep gee, geµ and geτ as free complex parameters

and fix the remaining ones (gµµ, gµτ , gττ ) by imposing the equality of the three elements

m22,m23 and m33 of the neutrino mass matrix Mν , written in terms of the parameters of

the ZB model in eq. (13), and in terms of the masses and mixings measured in neutrino

oscillation experiments in eq. (14), i.e.,

mij = (UDνUT )ij = ζfiaωabfjb, (39)

where we have defined ωab ≡ mag∗abmb, and ζ = µ

48π2M2 I(r), being r the ratio of the scalar

masses, r ≡ m2k/m

2h.

Because of the hierarchy among the charged lepton masses, me � mµ,mτ , it is natural to

assume that ωee, ωeµ, ωeτ � ωµµ, ωµτ , ωττ . Within the approximation ωea = 0, the equation

(39) for neutrino masses is simplified, and we can easily estimate the ranges of parameters

consistent with neutrino oscillation data. Thus in this section we neglect them, although we

keep all ωab in the full numerical analysis6 We then have

m22 ' ζf 2µτωττ , m23 ' −ζf 2

µτωµτ , m33 ' ζf 2µτωµµ. (40)

From the large atmospheric angle we expect

|ωττ | ' |ωµτ | ' |ωµµ|, (41)

which leads to a definite hierarchy among the corresponding gab couplings:

gττ : gµτ : gµµ ∼ m2µ/m

2τ : mµ/mτ : 1. (42)

It is now convenient to write the mass matrix elements mij in terms of the neutrino

masses and mixings. In the normal hierarchy case this gives

ζf 2µτωττ ' m3c

213s

223 +m2e

iφ(c12c23 − eiδs12s13s23)2 ,

ζf 2µτωµτ ' −m3c

213c23s23 +m2e

iφ(c12s23 + eiδc23s12s13)(c12c23 − eiδs12s13s23) ,

ζf 2µτωµµ ' m3c

213c

223 +m2e

iφ(c12s23 + eiδc23s12s13)2 , (43)

which for m3 ' 0.05 eV and m2 ' 0.009 eV, leads to

ζf 2µτ |ωab| ' 0.025 eV , a, b = µ, τ, (44)

6 We find that, in general, this is a very good approximation.

21

in agreement with the expectations of eq. (41).

In the inverted hierarchy case, eqs. (40) read

ζf 2µτωττ ' m1(c23s12 + eiδc12s13s23)2 +m2e

iφ(c12c23 − eiδs12s13s23)2 ,

ζf 2µτωµτ ' m1(s12s23 − eiδc12c23s13)(c23s12 + eiδc12s13s23)

+ m2eiφ(c12s23 + eiδc23s12s13)(c12c23 − eiδs12s13s23) , (45)

ζf 2µτωµµ ' m1(s12s23 − eiδc12c23s13)2 +m2e

iφ(c12s23 + eiδc23s12s13)2,

where m1 ' m2 ' 0.05 eV. It is important to notice that for eiφ ∼ eiδ ∼ 1 the matrix

elements mij are of the same order as in the NH spectrum, i.e.,

ζf 2µτ |ωab| ' 0.025 eV , a, b = µ, τ. (46)

and therefore the hierarchy of couplings in eq. (42) is also obtained. However, in the IH

case there is a strong cancellation for Majorana phases close to π, so we can obtain smaller

values of ωab. In particular, for φ = δ = π and the best fit values of the masses and mixing

angles we find

ζf 2µτ |ωµµ| ' 0.003 eV, (47)

which allows for a smaller gµµ and, as a consequence, a lighter mk still consistent with the

experimental limits. On the contrary, if φ ∼ π and δ ∼ 0, |ωττ | can be very small and

therefore gττ � (m2µ/mτ )

2 gµµ, although this cancellation has no phenomenological impact.

Therefore, although in the following analytic approximations we assume the hierarchy of

couplings in eq. (42), one has to keep in mind that a larger parameter space is expected to

be allowed when φ ' δ ' π. Indeed we will confirm in the full numerical analysis of section

V that this region is specially favoured for light mk.

Now we can estimate the lowest scalar masses able to reproduce current neutrino data.

Using the neutrino mass equation we can write7

m33

0.05 eV' 500|gµµ||fµτ |2

µ

M

TeV

MI(r). (48)

The upper bound on τ → 3µ decay implies that |gµµ| <∼ 0.4(mk/TeV), while the new MEG

limits on µ → eγ lead to ε|fµτ |2 <∼ 1.3 · 10−3(mh/TeV)2, where ε ≡ |feτ/fµτ | ∼ 1/2 (4) for

7 Notice that similar limits are derived from any of the 23 block elements ofMν when assuming the hierarchy

of the g couplings given in eq. (42).

22

NH (IH). Substituting these constraints in eq. (48) we obtain

m33

0.05 eV<∼ 0.26

µmk

εM2

( mh

TeV

)2

I(r), (49)

which can be translated into a lower bound on the scalar masses. Using that m33 ∼ 0.025

eV from neutrino oscillation data, if mh > mk then µ ≤ κmk and I(r) ∼ 1, so eq. (49)

implies that

mh > mk>∼

1 TeV√κ

NH, (50)

mh > mk>∼

3 TeV√κ

IH. (51)

On the contrary, if mh < mk, we find

mk > mh>∼√

mk

mh κ I(r)1 TeV NH, (52)

mk > mh>∼√

mk

mh κ I(r)3 TeV IH. (53)

From the above results8, we conclude that:

1. It is easier to reconcile an enhanced Higgs diphoton decay rate with neutrino oscillation

data if the former is due to the doubly charged scalar loop contribution, since the lower

bounds from neutrino masses are similar, while the BR(H → γγ) can be accounted

for by a heavier mk. Moreover, if the enhancement is due to a light mh, then mk can

not be very heavy, because otherwise neutrino masses are too small.

2. For a NH neutrino mass spectrum, it is possible to fit simultaneously neutrino oscilla-

tion data, lepton flavour violation constraints and an enhanced BR(H → γγ) only if

the trilinear coupling µ is large, namely κ >∼ 4(10) for min(mh,mk) = 500 (300) GeV,

respectively.

3. In general, the case of IH neutrino masses is in conflict with an enhanced Higgs dipho-

ton rate unless κ ∼ O(30). However if we take into account the strong cancellations

in ωµµ when φ ' δ ' π, and allow for a smaller m33 ∼ 0.003 eV, it is also possible to

fit all data with κ ∼ 4.

8 Our limits in the IH case differ from those in [11]. We traced this difference to the fact that in the

estimates of [11] the perturbativity bound |gµµ| < 1 is imposed, but for low masses, mk < 2 TeV, such

bound is always satisfied, and the relevant bound is |gµµ| <∼ 0.4(mk/TeV), which depends on mk and

changes the scaling with ε, leading to a weaker lower bound on the charged scalar masses in our case. We

thank Martin Hirsch for discussions about this point.

23

V. NUMERICAL ANALYSIS

In order to explore exhaustively the highly multi-dimensional parameter space of the

ZB model, naive grid scans are completely inappropriate, the method of choice is re-

sorting to Monte Carlo driven Markov Chains (MCMC) that incorporate all the cur-

rent experimental information described in precedence. As parameters we will use

{s2ij,∆A,∆S, δ, φ, fµτ ,mh,mk, µ, gee, geµ, geτ}, and we allow them to vary within the ranges

showed in table IV.

Had we tried to use our MCMC to obtain a posteriori probability distribution functions

with a canonical Bayesian meaning, the choice of priors would have had a significant role.

Nevertheless, since our aim is to explore where in parameter space could the ZB model

adequately reproduce experimental data without weighting in the available parameter space

volume (that is, the “metric” in parameter space given by the priors), we will represent

instead profiles of highest likelihood (equivalently profiles of minimal χ2 ≡ −2 lnL with L

the likelihood) which, on the contrary, can be interpreted on a frequentist basis. This is

not a choice that we make because of the merits or demerits of either statistical school: our

goal remains to understand if and where the ZB “works well”, i.e. could fit experimental

data. The interpretation of the results/plots will be clear: they show the regions where the

model is in agreement with data without regard to their size when the remaining information

(parameters and observables) is marginalized over9. In this case, exploring the parameter

space in a uniform, logarithmic or other manner, in some given parameter will not affect our

results (only the computational efficiency required to reach them will be, of course, affected).

For the modelling of experimental data we typically resort to individual Gaussian like-

lihoods for measured quantities. Bounds are implemented through smooth likelihood func-

tions that include, piecewise, a constant and a Gaussian-like behaviour. For the sake of

clarity: if the experimental bound for a given observable O is BO[90%CL] at 90% CL (1.64σ in

one dimension), the χ2 contribution associated to the model prediction Oth for this observ-

9 Typically both approaches should converge to similar results when (experimental) information abounds;

in a study such as this one, if they differ, rather than sticking to one or the other, from the physical point

of view we would only conclude that the current experimental data is not yet sufficient to pin down or

exclude the model.

24

able is

χ2(Oth) =

0, Oth < BO[90%CL]/1.64,(

1.64Oth

BO[90%CL]

− 1

)2 (1.640.64

)2, Oth ≥ BO[90%CL]/1.64.

In this way we avoid imposing sharp stepwise bounds or half-Gaussian with best value at

zero that may penalize deviating from null predictions when this might not be supported by

experimental evidence (in particular when the number of bounds included in the analysis is

significant).

Simulations are done for both normal and inverted hierarchy. In each point of the pa-

rameter space we compute the full χ2, including all measurements and bounds. In the plots

we show the regions with the total ∆χ2 ≤ 6, which corresponds to 95% confidence levels

with two variables.

Parameter Allowed range

∆S (7.50± 0.19)× 10−5 eV2

∆A (2.45± 0.07)× 10−3eV2

sin2 θ12 0.30± 0.13

sin2 θ23 (0.42± 0.04) ∪ (0.60± 0.04)

sin2 θ13 0.023± 0.002

δ, φ [0, 2π]

arg(gee), arg(geµ), arg(geτ ) [0, 2π]

fµτ , |gee|, |geµ|, |geτ | [10−7, 5]

mh [100, 2× 103] GeV

mk [200, 2× 103] GeV

µ [1, 2κ× 103] GeV

Table IV: Allowed ranges for the parameter scan (Neutrino oscillation parameters are obtained

from [51–53]).

To compare our results with the analysis presented a few years ago by some of us [12]

some remarks are in order: first, here we have updated the experimental input on LFV and

neutrino oscillation parameters, as well as LHC direct searches. The new limits, in particular

on µ → eγ, tend to reduce the allowed regions but not dramatically. Especially important

is the determination of sin θ13: as shown in [12], already before its measurement the ZB

25

Figure 6: mh vs mk for NH (left) and IH (right) for different values of the perturbative parameter

κ = 1, 5, 4π (dark to light colours).

model predicted a large mixing angle θ13 in the case of IH spectrum, close to the previous

experimental upper limit, while for NH any value of θ13 below the bound was allowed. In

fact, a very small value of θ13 would have ruled out the IH possibility within the ZB model.

Second, although the scanning of parameters is performed like in [12], we have chosen here to

present results in terms of profiles of highest likelihood, which are insensitive to the volume

of the parameter space and the priors used to scan it. This allows us to explore regions

where parameters are fine tuned (after all, Yukawa couplings always require a certain degree

of fine tuning). This is important since, as we have seen, the model is highly constrained at

present and less conservative assumptions could exclude it before time, at least in the region

of low masses. Moreover, we focus only on the region of masses with phenomenological

interest (mh,k < 2 TeV) precisely to explore better the region of low masses.

In fig. 6 we depict the points allowed by neutrino oscillation data and all low energy

constraints in the plane (mh,mk) for the two mass orderings (NH and IH) and different

values of the fine-tuning parameter in eq. (9) (κ = 1 darker, κ = 5 dark, κ = 4π light). The

results of the numerical analysis imply that in general the indirect lower bounds on mh and

mk from neutrino oscillation data and low energy constraints are stronger than the current

limits from direct searches, except when cancellations occur for δ, φ ∼ π, especially in the

IH case, and/or when naturality assumptions on µ are relaxed, allowing for κ = 4π. In

table V we summarize the lower bounds on the scalar masses obtained for the three values

26

NH IH

κ 1 5 4π 1 5 4π

mh (GeV) 700 (1000) 300 (400) 200 (250) 220 (> 2000) 100 (1000) 100 (650)

mk (GeV) 700 (1100) 300 (450) 200 (250) 200 (> 2000) 200 (1000) 200 (550)

Table V: Lower bounds for the scalar masses for NH and IH and the naturality constraints

parametrized by the three values of κ. We present results for δ = π (δ = 0) (see figs. 6 and

7).

Figure 7: δ vs mk in NH (left) and IH (right).

of the naturality parameter κ, and two illustrative values of the Dirac phase, δ = 0, π. For

δ ∼ −π/2, as might be suggested by a recent analysis [51], the bounds are slightly weaker

than in the δ = 0 case (see fig. 7).

The correlation between the CP phase δ of the neutrino mixing matrix and the scalar

masses is illustrated in fig. 7, where we plot δ versus the doubly charged scalar mass, mk.10

Such correlation is especially relevant in the IH case, where scalar masses lower than ∼ 1

TeV are only allowed if δ ∼ π. A similar correlation with the phase φ was already found in

[12] for IH spectrum, so we do not show it here.

Regarding the singly charged scalar h±, the width of its decay modes (eν, µν, τν) is fixed

by the fia couplings to leptons (see for instance [11, 12] for the relevant formulae). Therefore,

10 The correlation of δ with mh is entirely analogous.

27

Figure 8: Branching ratios of the charged singlet h to eν, µν, τν splitting the two currently allowed

octants of θ23, θ23 < 45◦ (θ23 > 45◦) left (right). One can see the dependence on δ for the

NH spectrum in the µν and τν channels. The most significant change between octants is the

interchange of the µν and τν for the IH case. The bands are 95% C.L. regions.

after the measurement of θ13, present neutrino oscillation data determine completely the BRs

of h from eqs. (37) and (38), up to a residual dependence on the CP phase δ in the case

of NH spectrum. In this case, a very precise measurement of the branching ratios in the

µν or τν channels (probably in a next generation collider) will predict the CP phase δ, and

viceversa. We show the ranges attainable by the different BRs in fig. 8, as a function of

δ, splitting the two currently allowed octants of θ23. The most significant change between

octants is the interchange of the µν and τν for the IH case. Clearly, the best option to

discriminate between hierarchies is the eν channel.

An important point of the ZB model is that the doubly charged scalar can decay to two

singly charged scalars, which are difficult to detect at the LHC. However, in fig. 6 we see

that for a NH neutrino mass spectrum mh > 200 GeV, and the channel k → hh is closed for

mk < 400 GeV. Therefore, present bounds on mk from dilepton searches at LHC discussed

in II D apply. For the IH case, the k → hh channel is always open and can be dominant,

unless κ = 1, for which we obtain that it is closed in the region mk < 440 GeV. Thus in

general current direct bounds from LHC are weaker.

Let us now turn to the gab couplings. We find always gττ � gµτ , both for the NH

28

Figure 9: log |gµµ/gµτ | and log |gττ/gµτ | vs δ for NH (left) and IH (right). The horizontal red lines

represent the naive approximation in eq. (42).

and IH cases, in agreement with the analytic estimates in eq. (42); however the expected

ratio gµµ/gµτ ∼ mτ/mµ is only fulfilled for the NH spectrum, since in the IH case large

cancellations when the phases of the PMNS matrix U are δ ∼ φ ∼ π lead to smaller

gµµ � gµτ . This can be seen in fig. 9, where we show the ratios gττ/gµτ and gµµ/gµτ

obtained in the numerical simulation as a function of δ, together with the expectation based

on the analytic approximations, which is just a constant fixed by the charged lepton masses

29

Figure 10: log |gµµ| vs mk for NH (left) and log |gµτ | vs mk for IH (right).

(red horizontal line)11.

To set the absolute scale of the couplings we present in fig. 10 the value of the largest

couplings against mk, namely gµµ in the NH case, and gµτ in the IH case. We see that in

both cases the couplings are always in the range from 10−2 to 1 and therefore they tend to

dominate the decays of the k++.

Regarding the couplings gea, which are not determined by the neutrino mass matrix,

bounds from LFV charged lepton decays strongly constrain geτ and geµ to be less than

O(0.01), while gee can be larger, O(1). The constraint on |geegeµ| from µ→ 3e implies that

|geegeµ| < 2.3× 10−5 (mk/TeV)2 and it is illustrated in fig. 11.

Since the widths of the k±± leptonic decay modes are directly related to these couplings,

from the above results we can readily infer the corresponding BRs. We find that the prob-

ability of k → eµ, eτ, ττ is always negligible (even in the IH case, geµ can be at most 0.1

and only when δ ∼ π). For mk<∼ 400 GeV, and NH neutrino spectrum, BR(k → ee) +

BR(k → µµ) ∼ 1, since the k → hh decay channel is closed; therefore k±± can not evade

current LHC bounds on doubly-charged scalar searches and the limit mk > 310 GeV applies

(400 GeV if no signal is found at 8 TeV with 20 fb−1 [36]). In the same mk range, for IH

11 In the NH case there can also be cancellations with the geτ terms, which have been neglected in eq. (43),

that would allow much smaller values of gττ and gµτ , but those only occur for κ = 4π and in a tiny region

of the parameter space.

30

Figure 11: log |geµ| vs log |gee| for NH (left) and IH (right).

neutrino spectrum the BR(k → µτ) can also be significant and the channel k → hh is open

(unless κ = 1, for which it is only open for mk > 440 GeV), thus the present bound is

weaker.

When the upcoming LHC 13-14 TeV data is available, it is important to take into account

that the decay channel k → hh is open for mk>∼ 400 GeV, and can be dominant, so in this

mass range limits on doubly-charged scalars from dilepton searches will not apply to the ZB

model. On the contrary, if a doubly charged scalar were detected at LHC in any mass range,

neutrino oscillation data and low energy constraints are powerful enough to falsify the ZB

model to a large extent. For instance, we know that BR(k → eµ, eτ, ττ) are negligible for

any neutrino mass spectrum, while a sizeable BR(k → µτ) is only compatible with an IH

spectrum.

VI. CONCLUSIONS

We have analyzed the ZB model in the light of recent data: the measured neutrino mixing

angle θ13, limits from the rare decay µ→ eγ and LHC results. Although the model contains

many free parameters, neutrino oscillation data and low energy constraints are powerful

enough to rule out sizeable regions of the parameter space. A large source of uncertainty

comes from the mass scale of the new physics, which is unknown. Since we are interested

31

on possible signatures at the LHC, we present results for the masses of the extra scalar

fields below 2 TeV. Previous analyses [11, 12] have shown that larger mass scales are always

allowed, given the absence of significant deviations from the SM besides neutrino masses.

Even within this reduced scenario, there is still a free mass parameter, the trilinear cou-

pling between the charged scalars, µ, which remains mainly unconstrained. Naturality ar-

guments together with perturbativity and vacuum stability bounds, indicate that µ can not

be much larger than the physical scalar masses, mk,mh, but it is not possible to determine

a precise theoretical limit. Because the neutrino masses depend linearly on the parameter

µ, the ability of the model to accommodate all present data is quite sensitive to the upper

limit allowed for it, so we have considered three limiting values, µ < κmin(mk,mh), with

κ = 1, 5, 4π. Within the above ranges for the mass parameters of the ZB model, we have

performed an exhaustive numerical analysis using Monte Carlo Markov Chains (MCMC),

incorporating all the current experimental information available, both for NH and IH neu-

trino masses. The results of the analysis are presented in sec. V and summarized in figs. 6

– 11.

We have addressed the possibility that the slight excess in the Higgs diphoton decay

observed by the ATLAS collaboration is due to virtual loops of the extra charged scalars

of the ZB model, h± and k±±. Note that in the Zee-Babu model, as the new particles

are singlets, there is a negative correlation between H → γγ and H → γZ. Although a

similar study has been performed in [22], it was limited to the scalar sector parameters of

the model, and neutrino data, which we find crucial to determine the allowed charged scalar

masses, was not included in the analysis. In agreement with [22], we find that in order to

accommodate an enhanced H → γγ decay rate, large and negative λhH , λkH couplings are

needed, together with light scalar masses mh < 200 GeV, mk < 300 GeV. Such couplings

are in conflict with the stability of the potential, unless the self-couplings λh,k are pushed

close to the naive perturbative limit, ∼ 4π. As a consequence, even if vacuum stability

and perturbativity constraints are satisfied at the electroweak scale, RGE running leads to

non-perturbative couplings at scales not far from the electroweak scale, as shown in fig. 2.

When neutrino data and low energy constraints are taken into account, we still find

regions of the parameter space in which such enhancement is compatible with all current

experimental data; in particular, it seems easier if the enhancement is due to the doubly-

charged scalar loop contribution. As can be seen in fig. 6, in the NH case, the trilinear

32

coupling µ should be near its upper limit, while in the IH case lower masses can be achieved

in the region δ ∼ φ ∼ π due to cancellations.

Regarding LHC bounds on the doubly-charged scalar mass, they are largely dependent

on the BRs of the k±± decay modes, namely same sign leptons `±a `±b and h±h±. The leptonic

decay widths are controlled by the gab couplings to the right-handed leptons, which are in

principle unknown. By imposing that the measured neutrino mass matrix is reproduced,

within the approximation me = 0 one obtains analytically that gττ : gµτ : gµµ ∼ m2µ/m

2τ :

mµ/mτ : 1, while there is no information on the gea couplings. Our numerical analysis

confirms the above ratio of couplings in the case of NH, but for the IH spectrum there can be

large cancellations if the PMNS matrix phases δ, φ are close to π, leading to gττ � gµτ ∼ gµµ.

In both cases, geµ, geτ <∼ 0.1.

Moreover, in NH, if mk < 400 GeV for κ = 4π (mk < 600 GeV if κ = 5), mh < mk/2 is

ruled out, therefore the decay channel k → hh is kinematically closed and the LHC bounds

from doubly-charged scalar searches can not be evaded. In IH, however, for δ ∼ φ ∼ π the

k → hh channel is open unless κ = 1, while if δ is very different from π, indirect bounds on

mk set a much stronger constraint than direct LHC searches.

As a consequence, if the light neutrino spectrum is NH, k decays mainly to ee, µµ, and

the current bound from LHC is mk > 310 GeV, while if the spectrum is IH, k may also decay

to µτ and hh, so the present bound is weaker, about 200 GeV. Were a doubly-charged boson

discovered at LHC, the measurement of its leptonic BRs could rule out the ZB model, or

predict a definite neutrino mass spectrum. Conversely, if a CP phase δ is measured in future

neutrino oscillation experiments to be quite different from π together with an IH spectrum,

the mass of the charged scalars of the ZB model will be pushed up well outside the LHC

reach.

Note: During the final stages of this work we became aware of [54], where an analysis

of the Zee-Babu model was performed. Our bounds on the scalar masses are comparable

to theirs taking into account the slightly different procedures, in particular that they fix

the neutrino oscillation parameters to their best fit values and we allow them to vary in

their two sigma range. While in our work we focus on prospects for the LHC, in [54] the

possibility of detecting the doubly charged singlet in a future linear collider is studied.

33

Acknowledgments

We are thankful to the authors of [54] for sharing with us their work and for useful discus-

sions. We also thank Marcela Carena, Ian Low and Carlos Wagner for discussions. This work

has been partially supported by the European Union FP7 ITN INVISIBLES (Marie Curie

Actions, PITN- GA-2011- 289442), by the Spanish MINECO under grants FPA2011-23897,

FPA2011-29678, Consolider-Ingenio PAU (CSD2007-00060) and CPAN (CSD2007- 00042)

and by Generalitat Valenciana grants PROMETEO/2009/116 and PROMETEO/2009/128.

M.N. is supported by a postdoctoral fellowship of project CERN/FP/123580/2011 at CFTP

(PEst-OE/FIS/UI0777/2013), projects granted by Fundacao para a Ciencia e a Tecnologia

(Portugal), and partially funded by POCTI (FEDER). J.H.-G. is supported by the MINECO

under the FPU program. He would also like to acknowledge NORDITA for their hospitality

during the revision of the final version of this work, carried out during the “News in Neutrino

Physics” program.

Appendix A: RGES IN THE ZB MODEL

16π2βH =3

8

[(g2 + g′2)2 + 2g4

]− (3g′2 + 9g2)λH + 24λ2

H + λ2hH + λ2

kH − 6y4t + 12λHy

2t

16π2βh = 6g′4 − 12g′2λh + 20λ2h + 2λ2

hH + λ2hk

16π2βk = 96g′4 − 48g′2λk + 20λ2k + 2λ2

kH + λ2hk

16π2βhH = 3g′4 − (15

2g′2 +

9

2g2)λhH + 12λHλhH + 8λhλhH + 2λkHλhk + 4λ2

hH + 6λhHy2t

16π2βkH = 12g′4 − (51

2g′2 +

9

2g2)λkH + 12λHλkH + 8λkλkH + 2λhHλhk + 4λ2

kH + 6λkHy2t

16π2βhk = 48g′4 − 30g′2λhk + 4λkHλhH + 8λhλhk + 8λkλhk + 4λ2hk , (A1)

16π2βg′ =5

3

(41

10+ 1

)g′3

16π2βg = −19

6g3

16π2βg3 = −7g33 , (A2)

16π2βt = yt

{9

2y2t −

(17

12g′2 +

9

4g2 + 8g2

3

)}. (A3)

34

Here g3, g, g′ are the SM SU(3)C , SU(2)L and U(1)Y gauge couplings, respectively, and

we have neglected all the Yukawa couplings but the top quark Yukawa, yt. We have also

neglected the fab, gab couplings because for the range of singlet scalar masses that we consider

(≤ 2 TeV), they are are severely constrained by LFV and are much smaller than 1 except

for some corners of the parameter space where some of them could be order one. For the

analysis of the vacuum stability of the scalar potential fab, gab are subdominant, specially

in the region of large and negative mixed scalar couplings required to accommodate the

diphoton excess in Higgs decays. For smaller mixed scalar couplings, however, a more

detailed analysis including all Yukawa couplings and taking also into account leading two-

loop effects (as well as top quark mass uncertainties for the Higgs quartic coupling) should

be carried out, which is beyond the scope of this work.

Appendix B: LOOP FUNCTIONS FOR H → γγ AND H → Zγ

• Functions relevant for H → γγ:

A0(x) = −x+ x2 f

(1

x

)(B1)

A1/2(x) = 2x+ 2x(1− x) f

(1

x

)(B2)

A1(x) = −2− 3x− 3x(2− x) f

(1

x

)(B3)

• Functions relevant for H → Zγ:

A0(x, y) = I1(x, y) (B4)

A1/2(x, y) = I1(x, y)− I2(x, y) (B5)

A1(x, y) = 4(3− tan2 θw)I2(x, y) +[(1 + 2x−1) tan2 θw − (5 + 2x−1)

]I1(x, y) (B6)

where

I1(x, y) =xy

2(x− y)+

x2y2

2(x− y)2

[f(x−1)− f(y−1)

]+

x2y

(x− y)2

[g(x−1)− g(y−1)

](B7)

I2(x, y) = − xy

2(x− y)

[f(x−1)− f(y−1)

](B8)

and, for a Higgs mass below the kinematic threshold of the loop particle, mH < 2mi,

f(x) = arcsin2√x , (B9)

g(x) =√x−1 − 1 arcsin

√x . (B10)

35

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[53] G. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo, et al., Global analysis of neutrino

masses, mixings and phases: entering the era of leptonic CP violation searches, Phys.Rev.

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D86 (2012) 013012, [arXiv:1205.5254] [InSPIRE].

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40

JHEP07(2012)030

Published for SISSA by Springer

Received: April 4, 2012

Accepted: June 15, 2012

Published: July 5, 2012

On the nature of the fourth generation neutrino and

its implications

Alberto Aparici, Juan Herrero-Garcıa, Nuria Rius and Arcadi Santamaria

Departament de Fısica Teorica, and IFIC, Universidad de Valencia-CSIC

Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, Spain

E-mail: [email protected], [email protected], [email protected],

[email protected]

Abstract: We consider the neutrino sector of a Standard Model with four generations.

While the three light neutrinos can obtain their masses from a variety of mechanisms

with or without new neutral fermions, fourth-generation neutrinos need at least one new

relatively light right-handed neutrino. If lepton number is not conserved this neutrino must

have a Majorana mass term whose size depends on the underlying mechanism for lepton

number violation. Majorana masses for the fourth-generation neutrinos induce relative

large two-loop contributions to the light neutrino masses which could be even larger than

the cosmological bounds. This sets strong limits on the mass parameters and mixings of

the fourth-generation neutrinos.

Keywords: Beyond Standard Model, Neutrino Physics

ArXiv ePrint: 1204.1021

c© SISSA 2012 doi:10.1007/JHEP07(2012)030

JHEP07(2012)030

Contents

1 Introduction 1

2 Light neutrino masses 3

2.1 Dirac masses 3

2.2 Seesaw 3

2.2.1 Type I: fermionic singlets 3

2.2.2 Type II: scalar triplet 4

2.2.3 Type III: fermionic triplets 5

2.3 Others 5

2.4 Weinberg operator 6

3 Fourth-generation neutrino masses 6

4 Light neutrino masses induced by new generations 11

5 Phenomenological constraints 14

5.1 Direct searches 14

5.2 Lepton flavour violation 16

5.3 Universality tests 16

5.4 Light neutrino masses 17

5.5 Neutrinoless double beta decay (0ν2β) 19

5.6 Four generations and the Higgs boson 20

6 Conclusions 22

A A model for calculable right-handed neutrino masses 23

1 Introduction

In the framework of the Standard Model (SM), fermions are grouped into three families,

each containing a doublet of quarks and a doublet of leptons. The number of families is not

a constructive parameter of the theory, and it could well be four or more; for this reason,

the enlargement of the SM with new generations has been commonly considered [1], and it

has proven to help in dealing with several problems, such as the lack of CP violation in the

SM to explain the baryon asymmetry of the universe [2] or the structure of the leptonic

mass matrices [3]. The currently available SM observables, however, constrain quite tightly

the properties of such new families [4], and the global electroweak fits seem to disfavour

a scenario with more than five generations [5, 6]; maybe the most striking result against

the existence of additional families is the LEP measurement of the number of neutrinos

– 1 –

JHEP07(2012)030

at the Z peak, which forbids more than three light neutrinos [4], but even this can be

dodged if the neutrinos of the new generations are too heavy to be produced in Z decays.

All in all, the existence of new generations is not actually excluded, and it seems worth

being considered [1], even more now that the LHC is working and exploring the relevant

mass range.

On the other hand, right-handed neutrinos constitute a common new physics proposal,

usually linked to the generation of neutrino masses. This is particularly interesting nowa-

days, ever since we gathered compelling evidence that neutrinos do have masses, that they

lie well below the other fermions’ ones, and that their mixing patterns differ extraordinar-

ily from those of the quark sector (for a review on the matter of neutrino masses see, for

example, [7]). The most straightforward way to construct a mass term for the neutrinos

within the SM is just to rely on the Higgs mechanism, and so to write the corresponding

Yukawa couplings; for that aim, one needs some fermionic fields which carry no SM charge:

right-handed neutrinos. However, we do not know whether neutrinos are Dirac or Majo-

rana. If they are Dirac, the smallness of the neutrino mass scale remains unexplained, for

it would be just a product of the smallness of the corresponding Yukawa couplings. In

order to provide such an explanation, many models and mechanisms have been proposed:

in the so-called see-saw models, the lightness of the neutrino mass scale is a consequence

of the heaviness of another scale. For instance, this scale is the lepton-number-violating

(LNV) Majorana mass of the extra right-handed neutrinos in type I see-saw [8–11]. On the

other hand, radiative models propose that neutrino masses are originated via suppressed,

high-order processes [12–15]. Although some of these proposals do not require right-handed

neutrinos, for the sake of generality it is a good idea to consider their possible involvement

in the generation of neutrino masses.

In this work we aim to discuss the naturality of the various scenarios arising when

new generations and right-handed neutrinos are brought together. Several previous works

have considered such association, either explicitly, in order to provide a mechanism for

mass generation, or implicitly, when assuming Dirac neutrinos in their analyses [16–25].

We argue that unless a symmetry is invoked which separates the new family from the first

three, the coexistence of both Dirac and Majorana neutrinos is not stable under radiative

corrections and doesn’t seem natural [25, 26]. Furthermore, the presence of a fourth family

plus a right-handed Majorana neutrino triggers the generation of Majorana masses for the

light species through a well-known mechanism [16, 17, 23, 25, 27–29]; the upper bounds on

the light neutrino masses can thus be translated into bounds on the mixings with the new,

heavy generations.

This paper is structured as follows. In section 2 we start by reviewing the different

mechanisms which can provide light neutrino masses. In section 3 we discuss the natural-

ness of those mechanisms to generate the fourth-family neutrino mass and conclude that

at least one right-handed neutrino is needed. Assuming that light neutrinos are Majorana,

we use naturalness arguments to provide a lower bound on the Majorana mass of the right-

handed neutrino. In section 4 we consider a minimal four generation SM with only one

relatively light right-handed neutrino and Majorana masses for light neutrinos parametrized

by the Weinberg operator [30, 31]. We describe the radiative, two-loop contribution of the

– 2 –

JHEP07(2012)030

heavy fourth-family neutrinos to the light neutrino mass matrix. In section 5 we discuss

the phenomenological consequences of this minimal four-generation scenario with heavy

Majorana neutrinos (lepton flavour violation, universality bounds, light neutrino masses,

neutrinoless double beta decay,. . . ), and we conclude in section 6. Appendix A is devoted

to describe an explicit example in which a finite Majorana mass for the fourth-generation

right-handed neutrino is radiatively generated.

2 Light neutrino masses

The huge hierarchy between neutrino masses and those of all other fermions has triggered

the appearance of many different mechanisms to explain the lightness of neutrinos. Here

we briefly review some of these mechanisms, with special emphasis on the frameworks that

are able to explain neutrino masses including a fourth generation, which will be discussed

in the next section.

2.1 Dirac masses

If there are right-handed neutrinos and a conserved global symmetry (for instance B − L)

prevents them from having a Majorana mass, neutrinos are Dirac particles, as all other

fermions in the SM. However, in this scenario there is no explanation for the smallness

of neutrino masses, having to impose by hand extremely tiny Yukawa couplings, approxi-

mately 6 (11) orders of magnitude smaller than the electron (top) one. Therefore, although

in principle it is possible, a Dirac nature does not seem the most natural option for neutri-

nos (but see, for example, [3], for a proposal in this direction which avoids tiny Yukawas).

2.2 Seesaw

Seesaw models are minimal extensions of the SM which can naturally lead to tiny (Majo-

rana) neutrino masses, keeping the SM gauge symmetry, SU(3)C ⊗ SU(2)L ⊗ U(1)Y and

renormalizability, but giving up the (accidental) lepton number conservation of the SM.

Let’s explain briefly the different types to fix notation.

2.2.1 Type I: fermionic singlets

In type I see-saw [8–11, 32], n SM fermionic singlets with zero hypercharge are added to

the SM; these have the quantum numbers of right-handed neutrinos, and can be denoted

by νRi. Note that to explain neutrino data, which requires al least two massive neutrinos,

a minimum of two extra singlets are needed. Having no charges under the SM, Majorana

masses for right-handed neutrinos are allowed by gauge invariance, so the new terms in the

Lagrangian are:

LνR = i νRγµ∂µνR −

(1

2νc

RMνR + ` φ Y νR + H.c.

), (2.1)

where ` and φ are respectively the lepton and Higgs SM doublets, φ = i τ2 φ∗ with τ2, the

second Pauli matrix, acting on the SU(2)L indices, M is a n× n symmetric matrix, Y is a

general 3×n matrix and we have omitted flavour indices for simplicity. After spontaneous

– 3 –

JHEP07(2012)030

symmetry breaking (SSB), 〈φ〉 = vφ with vφ = 174 GeV, the neutrino mass terms are

given by

Lν mass = −1

2

(νL νc

R

) ( 0 mD

mTD M

) (νc

L

νR

)+ H.c. , (2.2)

where mD = Y vφ. The mass scale for right-handed neutrinos is in principle free, however

if M � mD, upon block-diagonalization one obtains n heavy leptons which are mainly

SM singlets, with masses ∼ M , and the well-known see-saw formula for the effective light

neutrino Majorana mass matrix,

mν ' −mDM−1mT

D , (2.3)

which naturally explains the smallness of light neutrino masses as a consequence of the

presence of heavy SM singlet leptons.

2.2.2 Type II: scalar triplet

The type II see-saw [33–37] only adds to the SM field content one scalar triplet with

hypercharge Y = 1 (we adopt the convention that Q = Y + T3) and assigns to it lepton

number L = −2. In the doublet representation of SU(2)L the triplet can be written as a

2× 2 matrix, whose components are

χ =

(χ+/√

2 χ++

χ0 −χ+/√

2

). (2.4)

Gauge invariance allows a Yukawa coupling of the scalar triplet to two lepton doublets,

Lχ =(

(Y †χ )αβ ˜αχ`β + H.c.

)− V (φ, χ) (2.5)

where Yχ is a symmetric matrix in flavour space, and ˜= iτ2 `c. The scalar potential has,

among others, the following terms:

V (φ, χ) = m2χ Tr[χχ†]−

(µ φ†χ†φ+ H.c.

)+ . . . (2.6)

The µ coupling violates lepton number explicitly, and it induces a vacuum expectation

value (VEV) for the triplet via the VEV of the doublet, even if mχ > 0. In the limit

mχ � vφ this VEV can be approximated by:

〈χ〉 ≡ vχ 'µv2

φ

m2χ

; (2.7)

then, the Yukawa couplings in equation (2.5) lead to a Majorana mass matrix for the

left-handed neutrinos

mν = 2Yχvχ = 2Yχµv2

φ

m2χ

. (2.8)

Neutrino masses are thus proportional to both Yχ and µ. Such dependence can be un-

derstood from the Lagrangian, since the breaking of lepton number L results from the

– 4 –

JHEP07(2012)030

simultaneous presence of the Yukawa and µ couplings. As long as m2χ is positive and large,

vχ will be small, in agreement with the constraints from the ρ parameter, vχ . 6 GeV [38].1

Moreover, the parameter µ, which has dimensions of mass, can be naturally small, because

in its absence lepton number is recovered, increasing the symmetry of the model.

2.2.3 Type III: fermionic triplets

In the type III see-saw model [40, 41], the SM is extended by fermion SU(2)L triplets

Σα with zero hypercharge. As in type I, at least two fermion triplets are needed to

have two non-vanishing light neutrino masses. We choose the spinors Σα to be right-

handed under Lorentz transformations and write them in SU(2) Cartesian components~Σα = (Σ1

α,Σ2α,Σ

3α). The Cartesian components can be written in terms of charge eigen-

states as usual

Σ+α =

1√2

(Σ1α − iΣ2

α) , Σ0α = Σ3

α , Σ−α =1√2

(Σ1α + iΣ2

α) , (2.9)

and the charged components can be further combined into negatively charged Dirac fermions

Eα = Σ−α +Σ+cα . Using standard four-component notation the new terms in the Lagrangian

are given by

LΣ = i ~ΣαγµDµ · ~Σα −

(1

2Mαβ

~Σcα · ~Σβ + Yαβ `α

(~τ · ~Σβ

)φ+ H.c.

), (2.10)

where Y is the Yukawa coupling of the fermion triplets to the SM lepton doublets and the

Higgs, and M their Majorana mass matrix, which can be chosen to be diagonal and real

in flavour space.

After SSB the neutrino mass matrix can be written as

Lν mass = −1

2

(νL Σ0c

) ( 0 mD

mTD M

) (νc

L

Σ0

)+ H.c. , (2.11)

which is the same as in the type I see-saw just replacing the singlet right-handed neutri-

nos by the neutral component of the triplets, Σ0α, and therefore leads to a light neutrino

Majorana mass matrix

mν ' −mDM−1mT

D . (2.12)

However, since the triplet has also charged components with the same Majorana mass, in

this case there are stringent lower bounds on the new mass scale, M & 100 GeV.

2.3 Others

Here we briefly summarize non minimal mechanisms which also lead to Majorana light

neutrino masses. Most of these models do not include right-handed neutrinos and are

designed to obtain tiny Majorana masses for the left-handed SM neutrinos, so we can

anticipate that they will not be appropriate for the fourth generation.

1This bound is calculated after the inclusion of the one-loop corrections to the ρ parameter, and is

slightly looser than other previously obtained from electroweak global fits (see, for example, [39]). Note

also that the authors of [38] use a different normalisation for the VEV, and hence the difference between

their value and the one we present here.

– 5 –

JHEP07(2012)030

a) Radiative mechanisms. Small Majorana neutrino masses may also be induced by

radiative corrections [12–15]. Typically, on top of loop factors of at least 1/(4π)2,

there are additional suppressions due to couplings or ratios of masses, leading to

the observed light neutrino masses with a new physics scale not far above the elec-

troweak one.

b) Supersymmetry. There is an intrinsically supersymmetric way of breaking lepton

number by breaking the so-called R parity [42–51] (for a review see [52]). In this

scenario, the SM doublet neutrinos mix with the neutralinos, i.e., the supersymmet-

ric (fermionic) partners of the neutral gauge and Higgs bosons. As a consequence,

Majorana masses for neutrinos (generated at tree level and at one loop) are naturally

small because they are proportional to the small R-parity-breaking parameters.

2.4 Weinberg operator

As we have mentioned, it does not seem very natural that neutrinos are Dirac particles;

assuming that they are Majorana, we will be often interested in abstracting from the actual

mechanism of mass generation. In such case, if the light degrees of freedom are those of

the SM we can parametrise the Majorana masses in terms of the well-known dimension 5

Weinberg operator2 [30, 31]:

L5 =1

2

cαβΛW

(`αφ) (φ† ˜β) + H.c. , (2.13)

where ΛW � vφ is the scale of new physics and cαβ are model-dependent coefficients with

flavour structure, which in some models can carry additional suppression due to loop factors

(as is the case in radiative mechanisms) and/or ratios of mass parameters (for instance in

type II see-saw c ∝ µ/mχ). In those cases we will assume that ΛW is directly related to

the masses of the new particles and absorb all suppression factors in cαβ.

Upon electroweak symmetry breaking, the Weinberg operator leads to a Majorana

mass matrix for the light neutrinos of the form

mν = cv2φ

ΛW. (2.14)

Notice that if cαβ is suppressed, the scale ΛW does not need to be extremely large in order

to fit light neutrino masses and, thus, the Weinberg operator can parametrize a variety of

Majorana neutrino mass models, including those with masses generated radiatively.

3 Fourth-generation neutrino masses

If there exists a fourth generation, the fourth-generation neutrinos must be massive (with

masses & mZ/2 in order to avoid the strong limits for the number of active neutrinos

found at LEP). In principle all mass mechanisms available for the light neutrinos are also

available to the fourth-generation neutrinos, however, the fact that they must be quite

2In supersymmetric models, φ = Hu, since there are two Higgs doublets.

– 6 –

JHEP07(2012)030

massive changes completely the discussion of the naturalness of the different mechanisms.

Let us discuss them:

a) Dirac masses.

Since the fourth generation must be at the electroweak scale, this mechanism of

mass generation is quite natural for the fourth-generation neutrinos as long as lepton

number is conserved.

b) Fermionic singlets with Majorana mass.

If lepton number is not conserved there is no reason to forbid a Majorana mass term

for right-handed neutrinos (see the discussion below). However if mR � mD the

see-saw formula applies and the spectrum contains a relatively light, almost-active

neutrino with mass m4 ∼ m2D/mR, which must be heavier than mZ/2. Therefore

mR < m2D/mZ and the mass of the right-handed neutrino cannot be much larger

than the electroweak scale. On the other hand, if mR � mD there are two almost

degenerate neutrinos and we are in the pseudo-Dirac limit, which does not pose any

problem.

c) Scalar triplet.

In principle, as in the case of light neutrinos, scalar triplets could also be used to

obtain Majorana masses for the fourth-generation neutrinos. However, the strong

limits on the triplet’s VEV coming from the ρ parameter vχ . 6 GeV will yield

fourth-generation neutrino masses too small. This limit could be relaxed a bit if

radiative corrections to the ρ parameter coming from triplet masses are large and

such that cancel in part the deviations induced by the triplet’s VEV, but this will

require quite a high degree of fine tuning among rather different quantities. Therefore,

this mechanism alone is not a natural mechanism for the fourth-generation neutrino

masses.

d) Fermionic triplets.

This is similar to b), but together with the right-handed neutrinos there come new

charged fermions degenerate with the neutral component. Since production limits tell

us that the charged fermions must be heavier than about 100 GeV, in this case the

pseudo-Dirac limit is not possible. Moreover these new fermions cannot be extremely

heavy, because otherwise the active neutrino will be too light. We conclude that this

mechanism is viable but much more constrained than b).

e) Radiative mechanisms and SUSY with broken R parity.

Neutrino masses in these models are strongly suppressed with respect to the elec-

troweak scale by either loop factors, couplings and/or ratios of masses. Therefore

they are not viable for the fourth generation.

f) Weinberg operator.

In principle the Weinberg operator could also be used to give Majorana masses to

the fourth-generation neutrinos. However, it will provide masses O(v2φ/ΛW ) which

– 7 –

JHEP07(2012)030

νR4

νR4ℓi

νRk

νRk

ℓj

φ

φ

Figure 1. The two-loop process that provides a Majorana mass for the fourth-generation right-

handed neutrino in the framework of type I see-saw. The indices i and j represent any of the

four families; the index k, however, represents only the right-handed neutrinos associated to the

generation of masses for the light families, k = 1, 2, 3.

should be & mZ/2, so the scale of new physics ΛW can not be much larger than

the electroweak scale vφ and the effective theory does not make sense. Therefore the

Weinberg operator does not provide a useful parametrization of the fourth-generation

neutrino mass.

We can therefore conclude that only a), b) (which includes a) in some limit) and

possibly d) are good mechanisms for the fourth-generation neutrino masses. It seems then

that to describe correctly the fourth-generation neutrino one needs at least one right-

handed neutrino (either SM singlet or triplet) which has standard Dirac couplings to the

doublets. If this RH neutrino is a SM triplet, we have seen that its Majorana mass is in

the range 100 GeV . mR . few TeV.

However, if the right-handed neutrino is a SM singlet it could have a very small or

even vanishing Majorana mass term. Is it natural to have Dirac neutrinos for the fourth

generation? The answer is simple: yes, provided there is a symmetry that protects them

from acquiring a Majorana mass term. This is not the situation if the light neutrinos are

Majorana, as most of the SM extensions that we considered in section 2, and they can mix

freely with the heavy fourth family. We argue that in such a case a Majorana mass term

for the fourth right-handed neutrino should be allowed just on symmetry grounds, and in

fact, based on naturality arguments, a lower bound for this Majorana mass can be given.

Let us consider first the case in which the three light neutrinos obtain their masses via a

type-I see-saw containing heavy right-handed neutrinos with masses mRk (with k = 1, 2, 3)

of the order of 1012–1015 GeV. Since lepton number is not conserved it is natural to

consider a Majorana mass term for the fourth right-handed neutrino, νR4. However, in

order to satisfy the LEP bounds on the number of light active neutrinos, mR4 should be, at

most, of the order of a few TeV and, therefore, much smaller than mRk. Thus, one might

think that perhaps it is more natural to set directly mR4 = 0 and consider only Dirac

neutrinos for the fourth generation. The question that arises then is whether this choice

is stable or not under radiative corrections and what is the natural size one might expect

for mR4, since setting mR4 = 0 does not increase the symmetries of the Lagrangian. The

– 8 –

JHEP07(2012)030

answer can be obtained from the diagram in figure 1 which gives a logarithmically divergent

contribution to mR4 induced by the presence of the three heavy Majorana neutrino masses,

mRk [25, 26]. Thus, above the mRk scale, mR4 and mRk mix under renormalization and

do not run independently. Therefore, even if one finds a model in which mR4 = 0 at some

scale ΛC > mRk, mR4 will be generated by running from ΛC to mRk. This running can

easily be estimated from the diagram in figure 1 and, barring accidental cancellations, one

should require

mR4 &1

(4π)4

∑ijk

Yi4Y∗ikmRkY

∗jkYj4 ln(ΛC/mRk) &

1

(4π)4

∑ijk

Yi4Y∗ikmRkY

∗jkYj4 , (3.1)

where i, j = 1, 2, 3, 4, k = 1, 2, 3 and in the last step we have taken ln(ΛC/mRk) & 1.

Of course, given a particular renormalizable model yielding mR4 = 0 at tree level (see

appendix A for an explicit example) one should be able to compute the full two-loop

mass mR4, which will be finite and will contain the logarithmic contributions we have just

discussed.

Eq. (3.1) sets the lower bound that we had announced. Let us now estimate its

value; bearing in mind that in type I see-saw the light neutrino masses are given by3

(mν)ij ∼∑

k YikYjkv2φ/mRk. Then, by taking all mRk of the same order we can rewrite the

bound as

mR4 &∑ij

Yi4(m∗ν)ijYj4(4π)4

m2Rk

v2φ

. (3.2)

To give a conservative estimate we consider only the contribution of the first three gen-

erations because we expect their Yukawa couplings to the fourth right-handed neutrino

to be somewhat suppressed due to universality and LFV constraints [16, 17, 23, 53] (say,

Yk4 ∼ 10−2). Once we fix the neutrino masses and the Yukawa couplings between the

fourth-generation neutrino and the first three, mR4 grows quadratically with mRk. For

mν = 0.01 eV and Yk4 = 0.01 we obtain that mR4 is of order keV, GeV, PeV for

mRk = 109, 1012, 1015 GeV, respectively. The contribution of the fourth active neutrino is

not necessarily suppressed by the Yuwawa couplings and, in principle, by using it, even

more restrictive bounds on mR4 could be set. However, as (mν)44 is model-dependent,4 we

keep the most conservative bound.

Let us consider now the case in which the three light neutrinos obtain their masses

through the type II see-saw mechanism (see section 2.2.2), i.e., through their coupling to

a scalar triplet, χ, which develops a VEV. As discussed in section 3, this triplet cannot

be the only source Majorana masses for the fourth-generation neutrinos and, at least, one

right-handed neutrino is needed. We will assume then that there is a right-handed neutrino

which has Yukawa couplings to the four SM doublets. In this scenario one can easily see

that the right-handed neutrino will acquire, at two loops (as seen in figure 2), a Majorana

3Notice that by integrating out the three heavy right-handed neutrinos we obtain a Majorana neutrino

mass matrix for the four active neutrinos which is of the order of the light neutrino masses.4In this case, (mν)44 is the see-saw mass induced by only the three heavy right-handed neutrinos, thus,

it could even be zero if the Yukawas between the fourth lepton doublet and the three right-handed neutrinos

vanish for some reason.

– 9 –

JHEP07(2012)030

νR4 νR4ℓi

χ

ℓj

φ φ

Figure 2. The process that provides a Majorana mass for the right-handed neutrino associated to

the fourth generation in the framework of type II see-saw.

mass. This just reflects the fact that the right-handed neutrino mass mR4 and the trilinear

coupling of the triplet, µ, mix under renormalization. Applying the same arguments used

in the case of see-saw type I for light neutrino masses and the estimate of the diagram in

figure 2 we can write

mR4 &µ

(4π)4

∑ij

Yi4(Y ∗χ )ijYj4 , (3.3)

where Yχ are the Yukawa couplings of the triplet to the lepton doublets and, as before, we

have taken ln(ΛC/mχ) & 1. As in the type I see-saw case the result can also be expressed

in terms of the light neutrino masses (mν)ij ∼ (Yχ)ijµv2φ/m

2χ; thus

mR4 &∑ij


m2χ

v2φ

, (3.4)

which shows a similar structure to that obtained for type I see-saw, eq. (3.2). The same

result is obtained for type III see-saw, whose couplings are analogous to those of type I.

The similarity of the two results suggests that bounds of this type are quite general

and should appear in all kinds of four-generation models with light Majorana neutrinos.

In fact, as discussed in section 2.4, light Majorana neutrino masses can be parametrized in

many models by means of the Weinberg operator, eq. (2.13), which yields neutrino masses

given by eq. (2.14). Then, one could draw a two-loop diagram analogous to the diagrams

in figures 1 and 2 but with the propagators of heavy particles pinched and substituted

by one insertion of the Weinberg operator. This diagram is quadratically divergent and,

therefore, its contribution to mR4 can not be reliably computed in the effective field theory

because it depends on the details of the matching with the full theory from which the

effective one originates (in fact it vanishes in dimensional regularization or in any other

regularization scheme allowing symmetric integration), but one can use naive dimensional

analysis to estimate contributions of order

mR4 ∼ΛW

(4π)4

∑ij

Yi4c∗ijYj4 ∼


Λ2W

v2φ

(3.5)

which is precisely the result obtained in the see-saw models discussed above if one identifies

ΛW ∼ mRk,mχ. However, it is important to remark that in the low energy effective theory

mR4 is a free parameter, and eq. (3.5) is only a naive dimensional analysis estimate of what

one would expect in a more complete theory.

– 10 –

JHEP07(2012)030

4 Light neutrino masses induced by new generations

After the discussion above, to describe correctly the neutrino sector of models with four

generations we need just one relatively light right-handed neutrino, νR, to give Dirac mass

terms to the fourth-generation neutrinos, while Majorana masses for light neutrinos can

be parametrized by the Weinberg operator. We will work in this minimal four-generation

scenario, thus, the relevant part of the Lagrangian for our discussion is

LY = −¯YeeRφ− ¯yνRφ−1

2νc

RmRνR +1

2v2φ

(`φ)mL(φ† ˜) + H.c. , (4.1)

where ` and eR contain the four generation components while νR is the only right-handed

neutrino. Thus Ye is a completely general 4×4 complex matrix, y is a 4 component column

vector, mR is just a number and mL is a general complex symmetric 4 × 4 matrix. The

Dirac limit is recovered when mR = 0 and mL = 0. Since light neutrino masses are very

small, we will assume mL � vφ while mR, as we immediately see, cannot be very large

to ensure there are only three light active neutrinos. Moreover, as shown in the previous

section, we do not expect it to be zero if mL is not zero.

Above we have taken for mL a general complex symmetric 4 × 4 matrix in spite of

the fact that to describe the light neutrino sector we just need a 3 × 3 matrix. This is

because in most of the neutrino mass models one also obtains contributions to the fourth-

generation Weinberg operator. For instance, we give below the values of mL one obtains

for the different types of seesaw.

If the three light neutrino masses are generated by the seesaw mechanism type I or

type III, we need three of the right-handed neutrinos much heavier than the fourth. We

can always choose a basis in which the Majorana mass matrix of right-handed neutrinos

is diagonal and integrate out the three heavy right-handed neutrinos. The result can be

writen in terms of the Weinberg operator in (4.1) with

(mL)αβ = −∑

k=1,2,3

(Yν)αk (Yν)βkmRk

v2φ , (4.2)

where mRk are the eigenvalues of the diagonal Majorana mass matrix of the three heavy

right-handed neutrinos, while (Yν)αk are the Yukawa couplings of the three heavy right-

handed neutrinos with the four lepton doublets. Then, the 4 × 4 mass matrix (4.2) is

projective and has at most rank 3.

If the three light neutrino masses are generated by the VEV of a triplet (type II

see-saw), we will have

(mL)αβ = 2(Yχ)αβvχ (4.3)

being (Yχ)αβ the Yukawa couplings of the 4 lepton doublets to the triplet and vχ ∼ µv2φ/m

2χ

its VEV. In this case, mL is a completely general 4× 4 symmetric complex matrix.

After SSB the neutrino mass matrix (in the basis (νcLα, νR)) is

M =

(mL yvφyTvφ mR

). (4.4)

– 11 –

JHEP07(2012)030

To diagonalize this mass matrix we perform first a 4×4 rotation in order to separate heavy

from light degrees of freedom, so we change from the flavour basis (νe, νµ, ντ , νE) to a new

basis ν ′1, ν′2, ν′3, ν′4 in which the first three states are light (with masses given by mL) and

only ν ′4 mixes with νR. Then, we have να =∑

i Vαiν′i (i = 1, · · · , 4, α = e, µ, τ, E), where

V is a orthogonal matrix, and we define

Nα ≡ Vα4 =yα√∑β y

2β

. (4.5)

Now, we are free to choose ν ′1, ν′2, ν′3 in any combination of νe, νµ, ντ , νE as long as they are

orthogonal to ν ′4, i.e.,∑

α VαkNα = 0 for k=1,2,3. The orthogonality of V almost fixes allits elements in terms of Nα, but still leaves us some freedom to set three of them to zero.Following [16, 17] we choose Vτ1 = VE1 = VE2 = 0 for convenience. The transpose of thematrix V is:

V T =

Nµ√N2e +N2

µ

−Ne√N2e +N2

µ

0 0

NeNτ√(N2

e +N2µ)(1−N2

E)

NµNτ√(N2

e +N2µ)(1−N2

E)

−N2e −N2

µ√(N2

e +N2µ)(1−N2

E)0

NeNE√(1−N2

E)

NµNE√(1−N2

E)

NτNE√(1−N2

E)−√

(1−N2E)

Ne Nµ Nτ NE .

(4.6)

After this rotation the neutrino mass matrix is

M =

mL

ω1 0

ω2 0

ω3 0

ω1 ω2 ω3

0 0 0

ω4 mD

mD mR

, (4.7)

where (mL)kk′ = (V mLVT)kk′ is a 3 × 3 matrix with k, k′ = 1, 2, 3, ωk = (V mLV

T)4k,

ω4 = (V mLVT)44 and mD = vφ

√∑α y

2α. Since mL, ωk, ω4 � mR,mD, the matrix M

can be block-diagonalized using the see-saw formula. Then, the mass matrix of the light

neutrinos (at tree level) will be

m(0)ν = mL −

mR

mRω4 −m2D

~ω · ~ωT , (4.8)

while the heavy sector will be obtained after diagonalizing the 2× 2 matrix

MH =

(ω4 mD

mD mR

). (4.9)

Neglecting ω4, this diagonalization leads to two Majorana neutrinos

ν4 = i cos θ(−ν ′4 + ν ′c4 ) + i sin θ(νR − νcR) (4.10)

ν4 = − sin θ(ν ′4 + ν ′c4 ) + cos θ(νR + νcR) (4.11)

– 12 –

JHEP07(2012)030

with masses

m4,4 =1

2

(√mR

2 + 4m2D ∓mR

), (4.12)

and mixing angle tan2 θ = m4/m4. The imaginary unit factor i and the relative signs

in ν4 are necessary to keep the mass terms positive and preserve the canonical Majorana

condition ν4 = νc4. If mR � mD, we have m4 ≈ m4, tan θ ≈ 1, and we say we are in the

pseudo-Dirac limit while when mR � mD, m4 ≈ m2D/mR and m4 ≈ mR, tan θ ≈ mD/mR

and we say we are in the see-saw limit.

Eq. (4.8) can be used as long as mRω4 − m2D is different from zero. However, we

expect mR to be below few TeV and ω4 below 1 eV. Therefore mRω4 � m2D unless mD is

very small but, in that case, the fourth-generation neutrinos will be too light. Thus, the

correction to the 3 × 3 neutrino mass matrix is projective (only one eigenvalue different

from zero) and it is naturally order m2L and, therefore, negligible.

Summarizing, there are two heavy neutrinos 4 and 4 (with a small pollution from mL

which can be neglected) and a tree-level mass matrix for the light neutrinos m(0)ν ' mL.

Therefore, neglecting the small ωi’s in eq. (4.7), the 5×5 unitary matrix which relates the

flavour with the mass eigenstate basis can be written as U = UH ·UL, being UH the rotation

in the heavy sector which diagonalizes the mass matrix MH in eq. (4.9) and UL given by

UL =

V

0

0

0

0

0 0 0 0 1

W

0 0

0 0

0 0

0 0 0

0 0 0

1 0

0 1

, (4.13)

where V rotates from the ν ′i basis to the flavour basis (see eq. (4.6)) and W is the matrix

which diagonalizes mL. Within this approximation, the mixing among the light and the

heavy sector, which we wish to constrain, depends on (UL)α4 = Vα4 = Nα.

Having fixed the tree-level neutrino mass spectrum and given the huge hierarchies

present we should consider the stability of the results against radiative corrections. One can

check that there are no rank-changing one-loop corrections to the neutrino mass matrices.

This result can be easily understood in the ν ′i basis that we defined before, since the light

neutrinos (ν ′1, ν′2, ν′3) are decoupled from the heavy sector, ν4, ν4, so there are not one-loop

diagrams involving the fourth-generation neutrinos with light ones as external legs.

However it has been shown [16, 17, 23, 27] that two-loop corrections induced by the

fourth-generation fermions can generate neutrino masses for the light neutrinos even if they

were not present at tree level, see figure 3. In the ν ′i basis the result reads (see [23] for

details)

(mν)(2)ij = − g4

m4W

mRm2D

∑α

VαiVα4m2α

∑β

VβjVβ4m2βIαβ , (4.14)

where the sums run over the charged leptons α, β = e, µ, τ, E while i, j = 1, 2, 3, and Iαβ is a

loop integral which was discussed in [23]. When mR = 0, (mν)(2)ij = 0, as it should, because

– 13 –

JHEP07(2012)030

νi

νj

eα

W

W

eβν4, ν 4

Figure 3. Two-loop diagram generating light neutrino masses in the presence of a Majorana fourth

generation.

in that case lepton number is conserved. Also when mD = 0 we obtain (mν)(2)ij = 0, since

then the right-handed neutrino decouples completely and lepton number is again conserved.

To see more clearly the structure of this mass matrix we can approximate me =

mµ = mτ = 0; then, since we have chosen Vτ1 = VE1 = VE2 = 0, the only non-vanishing

element in (mν)(2)ij is (mν)

(2)33 and it is proportional to V 2

E3N2Em

4EIEE . Therefore, the largest

contribution to (mν)(2) is given by:

(mν)(2)33 = − g4

m4W

N2E(N2

e +N2µ +N2

τ )mRm2Dm

4EIEE

≈ g4

2(4π)4(N2

e +N2µ +N2

τ )mRm2

Dm2E

m4W

lnmE

m4

, (4.15)

where in the last line we have used the approximated expression of the loop integral IEEin the case mE � m4,4 � mW for definiteness, but other mass relations lead to analogous

conclusions. Keeping all the charged lepton masses one can easily show that the eigenvalues

of the light neutrino mass matrix are proportional to m4µ, m

4τ , m

4E which gives a huge

hierarchy between neutrino masses. Therefore, as discussed in [23, 26], these radiative

corrections cannot explain by themselves the observed spectrum of masses and mixings,

although they lead to a strong constraint for this kind of SM extensions which we will

analyze in the next section.

5 Phenomenological constraints

5.1 Direct searches

Let us now discuss the constraints that several phenomenological tests impose on the

parameters of this minimal four-generation (4G) model. Direct searches for the new heavy

leptons can be used to set limits on the Yukawa couplings and the Majorana mass of the

νR. In the case of the heavy charged lepton, searches at LEP [4] yield mE > 100.8 GeV

(assuming it decays rapidly to νW ; a slightly poorer bound is obtained if the lepton is

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JHEP07(2012)030

Figure 4. The shaded region shows the allowed values for the Majorana and Dirac masses of

the heavy neutrinos given the LEP bound mN > 33.5 GeV on stable neutrinos with both Dirac

and Majorana masses. We also display a dashed line in the 62.1 GeV limit for unstable neutrinos.

Two more lines are drawn for completeness, giving an idea of the combination of parameters that

produces two possible, allowed masses for the lightest heavy neutrino.

long-lived and can be tracked inside the detectors), which can be immediately translated

into a bound on the corresponding Yukawa. For the heavy neutrinos, we can have different

bounds depending on their stability and the Dirac or Majorana character of their masses [4]:

stable neutrinos, for example (understood here as ‘stable enough to get out of the detectors

after production’), are only constrained by the requirement that they don’t show up in the

invisible decays of the Z boson. Unstable (visible) neutrinos get tighter bounds due to the

non-observation of their decay products. As we are not making any a priori assumption

about the neutrino mass structure, we will select here the most conservative from this set of

bounds; that corresponds to a stable neutrino with both Dirac and Majorana mass terms,

for which we demand mN > 33.5 GeV [54]. The weakest bound for an unstable neutrino,

which applies if it has again both Dirac and Majorana mass terms, will also be of use; we

need in that case mN > 62.1 GeV [20]. As these bounds apply to the physical masses of

the neutrinos, which as seen in eqs. (4.10)–(4.12) are nonlinear combinations of the Dirac

and Majorana components, we display in figure 4 the translation of the 33.5 and 62.1 GeV

bounds into the mD −mR plane, together with several other lines to give an idea of the

relations between physical masses and Lagrangian parameters.

As explained in section 4, the neutrino Yukawas yα encode the mixings between the

flavour-eigenstate neutrinos να and the mass eigenstates ν4,4. Thus, we can use mixing-

mediated LFV processes to constrain the values of the light neutrino Yukawas ye, yµ, yτ . It

is important to note, however, that the situation is not the same for ‘stable’ and unstable

neutrinos; so-called stable neutral leptons are constrained to decay outside the detectors,

which implies that the mean free path must go beyond O(m). The lightest of our heavy

– 15 –

JHEP07(2012)030

Experimental bounds at 90% C.L. [4, 56] Constraints on the mixings

B(µ→ eγ) < 2.4× 10−12 NeNµ < 2.85× 10−4

B(τ → eγ) < 1.85× 10−7 NeNτ < 0.079

B(τ → µγ) < 2.5× 10−7 NµNτ < 0.093

Table 1. Summary of the constraints derived from low-energy radiative decays.

neutrinos can only decay through mixing (the main channel being ν4 → `αW , α = e, µ, τ ,

with a possibly virtual W depending on the mass of the ν4), so this statement is actually

a constraint on the Yukawas, implying yα ∼ Nα . 10−6. This constraint is much stronger

than any other phenomenolgical bound, and so it ends the discussion for stable neutrinos,

which must have very small mixings that won’t be observable in low-energy experiments

in the near future (see below). For the rest of this section we will consider the case of

unstable neutrinos, which present a richer variety of constraints.

5.2 Lepton flavour violation

Let us now discuss the bounds on violation of lepton family number that can shed light

on the relevant mixings of our model; the most stringent limits are derived from the non-

observation of radiative decays of the form `α → `βγ.5 In our model, the ratios for such

processes are given by

B(`α → `βγ) ≡ Γ(`α → `βγ)

Γ(`α → `βνν)=

3α

2π

∣∣∣∣∣∣∑a=4,4

UβaU∗αaH (m2

a/m2W )

∣∣∣∣∣∣2

, (5.1)

where H(x) is a loop function that can be found in [53], and the sum proceeds over all the

heavy neutrinos6 (one in the Dirac case, two if they are Majorana). The weakest bounds

are obtained if only one neutrino with light mass runs inside the loop; this corresponds

either to the Dirac limit with a low mass or to a hard see-saw limit, with the heavy neutrino

almost decoupled due to its small mixing. We will assume this scenario in our calculations

in order to produce conservative bounds. Table 1 summarises the experimental limits and

the constraints that can be extracted from these processes.

5.3 Universality tests

A second class of constraints upon family mixing arises from the tests of universality in

weak interactions. For our purposes, these are either direct comparison of decay rates of

one particle into two different weak-mediated channels, or comparison of the decay rates

of two different particles into the same channel.7 If the weak couplings are to be the

5Bounds obtained from present data on µ–e conversion in nuclei [4] are of the same order. However, there

are plans to improve the sensitivity in µ–e conversion in 4 and even 6 orders of magnitude [55], therefore

we expect from this process much stronger bounds in the future.6Note this expression contains the contributions from the light neutrinos; by using unitarity of the mixing

matrix, they are included in the definition of H(x).7Data from neutrino oscillations can also be used to constrain the elements of the leptonic mixing

matrix [57], however, they lead to weaker bounds than the ones obtained here.

– 16 –

JHEP07(2012)030

Experimental bounds at 90% C.L. [4] Constraints on the mixings

Rπ→e/π→µ

RSMπ→e/π→µ

= 0.996± 0.005 N2e −N2

µ = 0.004± 0.005

Rτ→e/τ→µ

RSMτ→e/τ→µ

= 1.000± 0.007 N2e −N2

µ = 0.000± 0.007

Rτ→e/µ→e

RSMτ→e/µ→e

= 1.003± 0.007 N2µ −N2

τ = 0.003± 0.007

Rτ→µ/µ→e

RSMτ→µ/µ→e

= 1.001± 0.007 N2e −N2

τ = 0.001± 0.007

Table 2. Summary of the constraints derived from universality tests in weak decays. The ratios

marked as “SM” represent the theoretical predictions of a 3G Standard Model.

same for all families these rates should differ only in known kinematic factors or calculable

higher-order corrections. The relevant ratios are:

Rπ→e/π→µ ≡Γ(π → eν)

Γ(π → µν), Rτ→µ/τ→e ≡

Γ(τ → µνν)

Γ(τ → eνν),

Rτ→e/µ→e ≡Γ(τ → eνν)

Γ(µ→ eνν), Rτ→µ/µ→e ≡

Γ(τ → µνν)

Γ(µ→ eνν),

and their theoretical values in a 3G SM can be consulted, for example, in [58]. Comparison

of the experimental values and the 3G predictions yields values very close to 1, as can be

seen in table 2; in our 4G model, family mixing induces deviations from this behaviour

that must be kept under control. Essentially, these deviations result from the fact that

the flavour-eigenstate neutrinos νe, νµ, ντ have a small component of the heavy neutrinos

ν4, ν4, which cannot be produced in the decays of pions, taus or muons; the corresponding

mixings Ne, Nµ, Nτ are then forced to be small. In table 2 we also show the constraints

that this processes impose on the mixing parameters.

In figure 5 we collect all the relevant LFV constraints from tables 1 and 2. As can be

read from the graphs, the final bounds we can set on the mixings of the light families are

Ne < 0.08

Nµ < 0.03 (5.2)

Nτ < 0.3

5.4 Light neutrino masses

Finally, there is a further constraint that can be set upon the mixings of the model: as

explained in section 4, the two-loop mechanism which gives small Majorana masses for

the light neutrinos cannot explain by itself the observed pattern of masses in this simple

model; it, nevertheless, still has the potential to generate too large masses, which would

exclude the model. Of course, one could always invoke cancellations between these two-

loop masses and other contributions (for example, the Weinberg operator), but we think

– 17 –

JHEP07(2012)030

Figure 5. These two graphs present the allowed regions for the mixing parameters in our model

at 90% confidence level, according to several LFV tests. The left plot displays the constraints in

the Ne − Nµ plane, which are much more stringent and suffice to bound both Ne and Nµ. The

right plot displays the Ne − Nτ plane; the Nµ − Nτ plane offers slighly poorer constraints and is

not displayed.

that this wouldn’t be a natural situation and choose not to consider it. If we bar such

cancellations we need to impose that the two-loop masses don’t go above some limit, and

thus a bound can be set upon the parameters that participate in the two-loop mechanism,

essentially the mixings and the Majorana mass, as seen in equation (4.14). Figure 6 shows

the allowed regions for this constraint; the curves are constructed using the lowest possible

values of the fourth-generation Dirac masses, in order to provide conservative limits (this

implies using a different value of mD for each mR, as we must also impose that m4 is

above 62.1 GeV). We show two possible limits: m(2)ν < 0.05 eV ensures that the largest

two-loop mass is below the atmospheric mass scale; this, of course, does not guarantee

that it doesn’t distort the neutrino spectrum, which may contain smaller masses, so this

bound can be contemplated as rather conservative (particular models may need to impose

a stronger bound to be phenomenologically viable). An even more conservative constraint

is obtained if we impose m(2)ν < 0.3 eV, meaning that the largest of the two-loop masses

is not above the bound imposed by cosmology to each mass of the degenerate spectrum,∑kmk . 1 eV [59, 60]. Two-loop masses as large as 0.3 eV will in most cases spoil the

structure of neutrino masses, but there may be pathological cases in which such situation

is allowed (for example, if the Weinberg operator generates a massless neutrino and two

massive ones near the 0.3 eV limit; then the two-loop diagram might provide the third mass

to fit the mass splittings). Even with these conservative assumptions, the two-loop bound

proves to be much stronger than those derived from universality and LFV for most of the

parameter space. It is, therefore, a limit to be kept in mind when considering 4G models

with Majorana neutrinos.

– 18 –

JHEP07(2012)030

Figure 6. Summary of the constraints on the mixings of the model, as defined in equation (4.5).

The three horizontal lines present the upper bounds in equation (5.2), derived from universality

tests and limits on LFV processes. The two shaded areas display the allowed region derived from

the fact that the two-loop diagram does not disturb the correct structure for the light neutrino

masses, assumed to arise from any other mechanism. This last bound applies to any of the mixings.

5.5 Neutrinoless double beta decay (0ν2β)

In our framework, the contributions to the amplitude of neutrinoless double beta decay

(0ν2β) can be written as:

A = AL +Amd +A4 , (5.3)

where AL stands for the light neutrino contribution (i.e., neutrino masses mk � peff ∼100 MeV), given by

AL ∝light∑k

mkU2ekM

0ν2β(mk) ' meeM0ν2β(0) , (5.4)

with M0ν2β(0) ∝ 1/p2eff the nuclear matrix element. The cosmology upper bound on the

sum of neutrino masses,∑

kmk . 1 eV [59, 60], combined with neutrino oscillation data,

leads to an upper limit on each neutrino mass mk . 0.3 eV and on the element of the

neutrino mass matrix relevant to 0ν2β decay, mee . 0.3 eV.

Amd represents the additional, model dependent contribution due to the unknown

physics which generates the three light neutrino masses parametrized by the Weinberg

operator. We assume that this last term is negligible compared to AL, as it is the case

if the underlying mechanism for neutrino masses is any of the standard three see-saw

types [61].

– 19 –

JHEP07(2012)030

We focus then on the contribution from the fourth-generation neutrinos (ν4, ν4),

given by

A4 ∝ N2e

(m4 cos2 θM0ν2β(m4)−m4 sin2 θM0ν2β(m4)

)∝ N2

e

(cos2 θ

m4− sin2 θ

m4

)= N2

e

m24−m2

4

m4m4(m4 +m4)= N2

e

mR

m2D

, (5.5)

where we have used that M0ν2β(ma) ∝ 1/m2a for a = 4, 4, tan2 θ = m4/m4, m4m4 =

m2D and m4 − m4 = mR. From eqs. (5.4) and (5.5) we see that the fourth-generation

neutrino contribution to the 0ν2β amplitude can be dominant provided N2emR/m

2D >

mee/(100 MeV)2. Notice, in fact, that the value of mee could be zero if normal hierarchy

is realised and the neutrino phases have the appropriate values; in this extreme case the

only contribution would be that of the fourth generation, which would dominate 0ν2β.

Now we can exploit the dependence on N2emR of both A4 and (mν)

(2)33 in eq. (4.15)

to constrain the fourth-generation neutrino contribution to the 0ν2β decay amplitude,8

namely

A4 ≤(

4πmW

gmD

)4 2(mν)(2)33

m2E ln mE

m4

. 190(mν)(2)33

(50 GeV

mD

)4

GeV−1 , (5.6)

where we have taken into account the LEP limit, mE & 100 GeV and set ln(mE/m4) ' 1.

From this equation, it is clear that the largest fourth-generation contributions to the

amplitude A4 correspond to a small Dirac neutrino mass, mD. Imposing that the two-

loop mass matrix element (mν)(2)33 is below the cosmology upper bound, 0.3 eV, we obtain

A4 < 6×10−8(50 GeV/mD)4 GeV−1, while if we require that the two-loop contribution is at

most the atmospheric mass scale, 0.05 eV, we find A4 < 10−8(50 GeV/mD)4 GeV−1. On the

other hand, the non-observation of 0ν2β implies that A4 < 10−8 GeV−1 [62], while future

sensitivity is expected to improve this limit one order of magnitude. Bringing these two

results together, we see that, once the constraint from light neutrino masses is taken into

account, the contribution of the fourth-generation neutrinos to the 0ν2β decay amplitude

can reach observable values only in the small region of parameter space mD . 100 GeV

(see figure 4), even though it is the dominant one for a larger set of allowed masses and

mixings.

5.6 Four generations and the Higgs boson

It is well known that due to the presence of a new generation there is an enhancement of

the Higgs-gluon-gluon vertex, which arises from a triangle diagram with all quarks run-

ning in the loop. This vertex is enhanced approximately by a factor 3 in the presence

of a heavy fourth generation, therefore the Higgs production cross section through gluon

fusion at the LHC is enhanced by a factor of 9. However, Higgs decay channels are also

strongly modified, in particular the Higgs to gluon decays are equally enhanced, while

the γγ channel is reduced because of a cancellation between the quark and W contribu-

tions. Moreover some of these channels, γγ for instance, suffer from important electroweak

radiative corrections [63].

8Note that (mν)(2)33 receives contributions from Ne, Nµ and Nτ , while 0ν2β only involves Ne.

– 20 –

JHEP07(2012)030

With the first LHC data, ATLAS and CMS ruled out at 95% C.L the range 120 −600 GeV for a SM4 Higgs boson, assuming very large masses for the fourth-generation

particles [64]. However, different authors have noticed that if fourth-generation neutrinos

are light enough (mW /2 . mν4 . mW ), the decay mode of the Higgs into fourth-generation

neutrinos can be dominant for mH . 2mW [65–68]. Moreover, if the lightest fourth-

generation neutrino is long-lived this decay channel is invisible and the excluded range

for the SM4 Higgs boson is reduced to 160 − 500 GeV [67, 68]. In general, if the fourth-

generation neutrinos have both Dirac (mD) and Majorana (mR) masses, the Higgs can

decay to more channels: ν4ν4, ν4ν4 and ν4ν4.

Recently, ATLAS and CMS have analysed new data, including more Higgs decay chan-

nels [69, 70], and they have a preliminary low-mass (∼ 125 GeV) hint of the Higgs boson

in several channels.9 In particular there is an excess in the γγ channel with respect to

the SM3 prediction. For such a light Higgs, the expected ratio of number of events into

γγ for SM4 over SM3 is about 1.5 − 2.5 at leading order [72, 73]. However, a global

fit to all relevant observables (Higgs searches and electroweak precision data), assuming

Dirac neutrinos and a Higgs mass of 125 GeV, shows that data are better described by

the SM3 [74]. On the other hand, as commented above, within the SM4 the cancellations

in the γγ channel at leading order render next-to-leading order radiative corrections im-

portant. These corrections tend to decrease even further the two-photon production rate

σ(gg → H) × BR(H → γγ)|SM4. Therefore, were the 125 GeV Higgs hint confirmed, by

combining the γγ, ZZ∗, WW ∗ and the ff channels a perturbative SM4 with just one

SM Higgs doublet would be excluded, even in the case mν4 < mW [75, 76]. Otherwise,

in principle it seems possible that if mν4 . mW and ν4 is long-lived, some portion of the

low Higgs mass parameter space, previously allowed to be between 114− 160 GeV, is still

allowed by the new data. Moreover, if one does not trust the convergence of perturbation

theory in the γγ channel and drops it from the global analysis, including Higgs searches,

Rb and oblique parameters, the SM4 with Dirac neutrinos is strongly constrained but still

viable [77]. Considering neutrino Majorana masses will presumably open up even more the

allowed parameter space of the model.

The previous bounds from LEP on the masses of unstable (in collider sense) fourth-

generation neutrinos were mν4 > 62.1 GeV. Using CDF inclusive like-sign dilepton analysis,

ν4 masses below mW can be excluded for Higgs masses up to 2mW [67], therefore in this

case the ATLAS and CMS analysis for the Higgs boson still apply, and at least the range

120− 600 GeV for a SM4 Higgs boson is excluded.

To know definitely whether the SM4 Higgs boson is excluded or not, we will have to

wait for new data and a combined analysis of the different channels, γγ, ZZ∗, WW ∗ and ff ,

including correctly all radiative corrections. However, even if the SM-like four-generation

Higgs is excluded, many possibilities may arise in extensions of a four-generation scenario,

for instance, with an extra Higgs doublet (see [78, 79] where the observed signatures of

LHC are explained in the framework of 4G two-Higgs-doublet models).

9Also Fermilab CDF and D0 have presented some preliminary results pointing to some excess around

this mass which can be assigned mainly to H → bb decays in HW and HZ associated production [71].

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JHEP07(2012)030

6 Conclusions

We have addressed the question of the generation and nature of neutrino masses in the

context of the SM with four families of quarks and leptons.

The three light neutrinos can obtain their masses from a variety of mechanisms with

or without new neutral fermions, but the huge hierarchy among such masses and those

of the remaining fermions is more naturally explained assuming that they have Majorana

nature.

On the other hand, current bounds on fourth-generation neutrino masses imply that,

although in principle the same mechanisms are also available, most of them are not natural

or provide too small fourth-generation neutrino masses; therefore, we have argued that at

least one right-handed neutrino is needed. This would suggest that, contrary to the light

neutrinos, fourth-generation ones are naturally Dirac.

However, we have shown that if lepton number is not conserved in the light neutrino

sector, the right-handed neutrino must have a Majorana mass term whose size depends on

the underlying mechanism for LNV, unless Yukawa couplings of the light leptons to the

right-handed neutrino are forbidden. We have estimated the natural size of such Majorana

mass term within two frameworks for the light neutrino masses, namely see-saw type I

and type II. We have seen that, even if we set it to zero by hand in the Lagrangian at

tree level, it is generated at two-loops , and although it depends on the Yukawa couplings

and the LNV scale responsible for light neutrino masses, it can be up to the TeV scale.

We have developed a model where this Majorana mass is forbidden at tree level by a

global symmetry, and it is generated radiatively and finite once this symmetry is broken

spontaneously (see appendix A).

We have then considered a minimal four-generation scenario, with neutrino Majorana

masses parametrized by the Weinberg operator and one right-handed neutrino νR, which

has Yukawa couplings to the four lepton doublets and non-zero Majorana mass. We have

analyzed the phenomenological constraints on the parameter space of such a model, de-

rived from direct searches for four-generation leptons, universality tests, charged lepton

flavour-violating processes and neutrinoless double beta decay. We have pointed out that

the Majorana mass for the fourth-generation neutrino induces relatively large two-loop

contributions to the light neutrino masses, which can easily exceed the atmospheric scale

and the cosmological bounds. Indeed, this sets the strongest limits on the masses and

mixings of fourth-generation neutrinos, collected in figure 6.

To summarize, in the context of a SM with four generations, we have shown that if light

neutrinos are Majorana particles, it is natural that also the fourth-generation neutrino has

the Majorana character. We did so by calculating the fourth-neutrino Majorana masses

induced by the three light neutrino ones. This has important implications for the neutrino

and Higgs sectors of these models, which are being actively tested at the LHC.

– 22 –

JHEP07(2012)030

Acknowledgments

We thank Enrique Fernandez-Martınez and Jacobo Lopez-Pavon for fruitful discussions

on the matter of neutrinoless double beta decay. This work has been partially sup-

ported by the Spanish MICINN under grants FPA-2007-60323, FPA-2008-03373, FPA2011-

23897, FPA2011-29678-C02-01, Consolider-Ingenio PAU (CSD2007-00060) and CPAN

(CSD2007- 00042) and by Generalitat Valenciana grants PROMETEO/2009/116 and

PROMETEO/2009/128. A.A. and J.H.-G. are supported by the MICINN under the FPU

program.

A A model for calculable right-handed neutrino masses

In this appendix we present a model which gives a realistic pattern of neutrino masses in

the context of the SM with four-generations and in which the right-handed neutrino mass

of the fourth generation is generated radiatively and finite. This is an illustration of the

general (model-independent) mechanism discussed in section 3 which allowed us to estimate

the size of Majorana neutrino masses for the fourth-generation right-handed neutrinos if

the three light active neutrinos are Majorana particles.

Let us consider the SM with four generations and four right-handed neutrinos νRi

(i = 1, · · · , 4). To implement the ordinary see-saw, we need three of them very heavy

while one of them should be much lighter in order to avoid a too light fourth-generation

active neutrino. Then, it is natural to require that one of the fourth right-handed neutrino

is massless at tree level and let its mass be generated by radiative corrections. For that

purpose we add three extra chiral singlets sLa (a = 1, · · · , 3). In order to break lepton

number we will also include a complex scalar singlet σ

We assign lepton number in the following way

`j → eiα`j , eRj → eiαeRj , νRj → eiανRj , σ → eiασ ; (A.1)

the sLa do not carry lepton number. With these assignments and the requirement that

lepton number is conserved we have the following Yukawa Lagrangian

LY = −` YeeRφ− ` YννRφ− σ νR y∗ sL −

1

2sc

LM∗sL + H.c. , (A.2)

where Ye and Yν are the ordinary four-generation Yukawa couplings, yia, along this ap-

pendix, is a general 4× 3 matrix while M is a symmetric 3× 3 matrix, which without loss

of generality can be taken diagonal and positive. We choose the scalar potential in such a

way that lepton number is conserved and subsequently spontaneously broken by the VEV

of σ, vσ = 〈σ〉. Thus, the model will contain a singlet Majoron. Alternatively, we could

also choose to softly break lepton number in the potential to avoid the Majoron without

changing the point we would like to illustrate. Before spontaneous symmetry breaking

only sLa are massive. We will take M very large (around GUT scale). After σ gets a VEV

(which is somewhat free, but we can take it just a bit below M), we will have a mass

– 23 –

JHEP07(2012)030

νR4 νR4

sLa

sLb

ℓi νRj

νRk ℓl

〈σ〉

〈σ〉

φ

φ

Figure 7. The process which generates masses for the right-handed neutrinos at two loops.

matrix for the combined system νR − sL of see-saw type. Therefore, if y vσ �M the four

right-handed neutrinos will get a 4× 4 Majorana mass matrix

M(0)R ' v2

σ yM−1yT ; (A.3)

this is basically the see-saw formula but applied to the right-handed neutrinos and changing

the VEV of the Higgs doublet for that of the singlet σ. This matrix has rank 3 and,

therefore, only three of the right-handed neutrinos will obtain a tree-level mass. The other

neutrino will remain massless at tree level. However, at two loops, due to the mechanism

described in section 3, also the fourth right-handed neutrino will acquire a Majorana mass.

We depict the diagram giving rise to this mass in figure 7; the diagram is obviously finite

by power counting and the generated mass matrix can be estimated as

M(2)R ∼ v2

σ

(4π)4(Y †ν Yν)T yM−1yT Y †ν Yν ln

(M

yvσ

)(A.4)

Since Ye does not enter in these calculations we can choose a basis in which Ye is

arbitrary but Yν is diagonal and real. If we take the logarithm order 1, ln(Myvσ

)∼ 1, we

see that M(2)R is also projective but in a different direction, given by y′ = (Y †ν Yν)Ty; then

we can write the full right-handed neutrino mass matrix as

MR ∼ v2σ

(yM−1yT +

1

(4π)4y′M−1y′

T), (A.5)

which, in general, has rank 4 and gives a Majorana mass to the fourth right-handed neu-

trino. To see how it works, let us discuss a simplified example, with the following structure

for the sL Yukawas:

y =

y1 0 0

0 y2 0

0 0 y3

0 0 y4

. (A.6)

– 24 –

JHEP07(2012)030

Let us also choose M diagonal and with elements Mi; then, at tree level we obtain an

almost diagonal mass matrix,

M(0)R = v2

σ

y21

M10 0 0

0y22

M20 0

0 0y23

M3

y3y4

M3

0 0 y3y4

M3

y24

M3

, (A.7)

which has a zero eigenvalue. At two loops we will have

M(2)R =

v2σ

(4π)4

y′21M1

0 0 0

0y′22M2

0 0

0 0y′23M3

y′3y′4

M3

0 0y′3y

′4

M3

y′24M3

, (A.8)

with y′i = yi(Yν)i2, and (Yν)i the diagonal elements of Yν . M

(2)R has also rank 3. However,

the sum of M(0)R and M

(2)R has rank 4, and the fourth νR acquires a mass. We can estimate

it by considering M(2)R a small perturbation to M

(0)R and find that

mR4 ∼v2σ

(4π)4M3

y24 y

23

y23 + y2

4

((Yν)2

4 − (Yν)23

)2, (A.9)

while the mass of the third right-handed neutrino is of order (the other two are also order

y2v2σ/M as can be seen from the mass matrix)

mR3 ∼ (y23 + y2

4)v2σ

M3. (A.10)

Therefore, if we rewrite the fourth-generation right-handed neutrino mass mR4 in terms of

mR3 we have

mR4 ∼mR3

(4π)4

y24 y

23

(y23 + y2

4)2

((Yν)2

4 − (Yν)23

)2, (A.11)

which is roughly the structure that one would expect from the effective theory obtained

by integrating the new fermions sLa, i.e, mR4 obtains a contribution proportional to the

heavy right-handed Majorana masses mR3 suppressed by a two-loop factor and Yukawa

couplings. After all, the diagram in figure 7 reduces to the diagram in figure 1 when the

fermion lines of sLa are contracted to a point. The result also shows that, as expected,

the exact coefficient depends on the details of the model. These expressions could be

generalized to a more general structure of Yukawa couplings, leading to similar, although

more complicated expressions.

As for other features of this model, we will just mention that as lepton number is broken

spontaneously, a Majoron will appear. Since the Majoron is a singlet and vσ is large their

couplings to standard model particles are suppressed and, therefore, this Majoron should

not create any problem. On the other hand, it could have some advantages in cosmological

contexts; if lepton number is also broken softly (for instance with a mass term σ2) the

– 25 –

JHEP07(2012)030

Majoron will become a massive pseudo-Majoron, which could constitute a good dark matter

candidate.

In any case, this simple example illustrates how the general mechanism discussed in

section 3 works in a complete renormalizable model; if mR4 is zero at tree level and light

neutrinos are Majorana (therefore lepton number is not conserved), in general mR4 will be

generated at two-loops with the behaviour discussed in section 3.

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JHEP07(2011)122

Published for SISSA by Springer

Received: April 26, 2011

Revised: May 30, 2011

Accepted: June 27, 2011

Published: July 28, 2011

Neutrino masses from new generations

Alberto Aparici, Juan Herrero-Garcıa, Nuria Rius and Arcadi Santamaria

Depto. de Fısica Teorica, and IFIC, Universidad de Valencia-CSIC,

Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, Spain

E-mail: [email protected], [email protected], [email protected],

[email protected]

Abstract: We reconsider the possibility that Majorana masses for the three known neu-

trinos are generated radiatively by the presence of a fourth generation and one right-handed

neutrino with Yukawa couplings and a Majorana mass term. We find that the observed

light neutrino mass hierarchy is not compatible with low energy universality bounds in this

minimal scenario, but all present data can be accommodated with five generations and two

right-handed neutrinos. Within this framework, we explore the parameter space regions

which are currently allowed and could lead to observable effects in neutrinoless double beta

decay, µ–e conversion in nuclei and µ → eγ experiments. We also discuss the detection

prospects at LHC.

Keywords: Beyond Standard Model, Neutrino Physics


c© SISSA 2011 doi:10.1007/JHEP07(2011)122

JHEP07(2011)122

Contents

1 Introduction 1

2 Framework and review 3

3 Four generations 6

4 Five generation working example 9

4.1 The five generations model 9

4.2 Two-loop neutrino masses 10

4.2.1 Normal hierarchy 11

4.2.2 Inverted hierarchy 12

4.3 The parameters of the model 13

4.3.1 Lepton flavour violation processes (µ → eγ and µ–e conversion) 14

4.3.2 Universality bounds 16

4.3.3 Neutrinoless double beta decay (0ν2β) 18

5 Collider signatures 19

6 Summary and conclusions 21

A The neutrino mass two-loop integral 23

1 Introduction

Neutrino oscillation data [1–14] require at least two massive neutrinos with large mixing,

providing one of the strongest evidences of physics beyond the Standard Model (SM).

However, the new physics scale responsible for neutrino masses is largely unknown. With

the starting of the LHC, new physics scales of order TeV will become testable through

direct production of new particles, so it is very interesting to explore low-energy scenarios

for neutrino masses. Moreover, typically, these scenarios also lead to observable signatures

in precision experiments, such as violations of universality, charged lepton flavour violating

(LFV) rare decays such as ℓi → ℓjγ or µ–e conversion in nuclei, which, being complementary

to the LHC measurements, may help to discriminate among different models. Regarding

the fundamental question of the neutrino mass nature, Dirac or Majorana, lepton-number-

violating low-scale models may give additional contributions to neutrinoless double beta

(0ν2β) decay process, shedding new light on this issue.

On the other hand, one of the most natural extensions of the SM that has been

extensively explored in the last years is the addition of one (or more) sequential generations

of quarks and leptons [15]. This extension is very natural and has a rich phenomenology

– 1 –

JHEP07(2011)122

both at LHC as well as in LFV processes. Moreover, new generations address some of the

open questions in the SM and can accommodate emerging hints on new physics (see for

instance [16] for a recent review).

Theoretically, apart from simplicity, there are no compelling arguments in favour of

only three families. In theories with extra dimensions one can relate the number of families

to the topology of the compact extra dimensions or set constraints on the number of chiral

families and allowed gauge groups by requiring anomaly cancellation. Then, one can build

models to justify only three generations at low energies. However, one could also build other

models in order to justify four or more generations. In the SM in four dimensions anomalies

cancel within each generation and, therefore, the number of families is in principle free.

From the phenomenological point of view it seems that the most striking argument

against new generations is the measurement of the invisible Z-boson decay width, Γinv,

which effectively counts the number of light degrees of freedom coupled to the Z-boson

(lighter than mZ/2) which is very close to 3 [17]. However, if neutrinos from new families

are heavy they do not contribute to Γinv and, then, additional generations are allowed.

Still, pairs of virtual heavy fermions from new generations contribute to the electroweak

parameters and spoil the agreement of the SM with experiment. Global fits of models

with additional generations to the electroweak data have been performed [18, 19] and the

conclusion is that they favour no more that five generations with appropriate masses for

the new particles. Although some controversy exists on the interpretation of the data

(see for instance [20]) most of the fits make some simplifying assumptions on the mass

spectrum of the new generations and do not consider Majorana neutrino masses for the

new generations or the possibility of breaking dynamically the gauge symmetry via the

condensation of the new generations’ fermions; all these will give additional contributions

to the oblique parameters and will modify the fits. Therefore, in view that soon we will

see or exclude new generations thanks to the LHC, it is wise to approach this possibility

with an open mind.

From the discussion above, it seems that neutrinos from new generations are very

different from the ones discovered up to now, since they should have a mass 1011 times

larger. However, this apparent difference is naturally explained within the framework that

we are going to explore. In the SM neutrinos are massless because there are no right-handed

neutrinos and because, with the minimal Higgs sector, lepton number is automatically

conserved. We now know that neutrinos have masses, therefore the SM has to be modified

to accommodate them; the simplest possibility is to add three right-handed neutrinos with

Dirac mass terms, like for the rest of the fermions in the SM. If one then considers the

SM with four generations (and four right-handed neutrinos), it is very difficult to justify

why the neutrino from the fourth generation is 1011 times heavier than the three observed

ones. This difficulty is alleviated if right-handed neutrinos have Majorana masses at the

electroweak scale and the Dirac masses of the neutrinos are of the order of magnitude of

their corresponding charged leptons [21]. Then, the see-saw mechanism is operative and

gives neutrino masses m1 ∼ m2e/M , m2 ∼ m2

µ/M , m3 ∼ m2τ/M , m4 ∼ m2

E/M ∼ mE (we

denote by E the fourth generation charged lepton). Although with a common Majorana

mass M at the electroweak scale it is not possible to obtain m3 light enough to fit the

– 2 –

JHEP07(2011)122

observed neutrino masses, this could be solved by allowing different Majorana masses for

the different generations; but then one should explain why M2,M3 ≫ M4.

Right-handed neutrinos, however, do not have gauge charges and are not needed to

cancel anomalies, therefore their number is not linked to the number of generations. In fact,

an extension of the SM with four generations and just one right-handed neutrino with both

Dirac and a Majorana masses at the electroweak scale leads, at tree level, to three massless

and two heavy Majorana neutrinos. Since lepton number is broken in the model, the three

massless neutrinos acquire Majorana masses at two loops therefore providing a natural

explanation for the tiny masses of the three known neutrinos [22].1 More generally, it has

been shown that in the SM with nL lepton doublets, nH Higgs doublets and nR < nL right-

handed neutrino singlets with Yukawa and Majorana mass terms there are nL−nR massless

Majorana neutrinos at tree level, of which nL−nR−max(0, nL−nHnR) states acquire mass

by neutral Higgs exchange at one loop [24–26]. The remaining max(0, nL − nHnR) states

get masses at two loops. Similar extensions could be built with additional hyperchargeless

fermion triplets, like in type III see-saw.

In this work we reconsider the model of ref. [22], without enlarging the scalar sector of

the SM but allowing for extra generations. The paper is organised as follows. In section 2

we summarize current neutrino data and searches for new generations. In section 3 we

review the radiative neutrino mass generation at two loops, and show that the observed

light neutrino mass hierarchy can not be accommodated in the minimal scenario with

four generations. In section 4 we present a five generation example which leads to the

observed neutrino masses and (close to tribimaximal) mixing. We introduce a simple

parametrization of the model and explore the parameter space allowed by current neutrino

data, universality, charged lepton flavour violating rare decays ℓi → ℓjγ and 0ν2β decay, as

well as the regions that will be probed in near future experiments (MEG, µ–e conversion

in nuclei). Section 5 is devoted to collider phenomenology and we summarize our results

in section 6.

2 Framework and review

It has been well established in the last decade that neutrinos are massive, thanks to the re-

sults obtained with solar [1–4, 11], and atmospheric [6, 7, 10] neutrinos, confirmed in exper-

iments using man-made beams: neutrinos from nuclear reactors [5] and accelerators [8, 12].

The minimum description of all neutrino data requires mixing among the three neutrino

states with definite flavour (νe, νµ, ντ ), which can be expressed as quantum superpositions

of three massive states νi (i = 1, 2, 3) with masses mi. The standard parametrization of

the leptonic mixing matrix, UPMNS, is:

UPMNS =

c13c12 c13s12 s13e−iδ

−c23s12 − s23s13c12eiδ c23c12 − s23s13s12e

iδ s23c13

s23s12 − c23s13c12eiδ −s23c12 − c23s13s12e

iδ c23c13

eiφ1

eiφ2

1

, (2.1)

1Two-loop quantum corrections within the SM with only two massive Majorana neutrinos also lead to

a (tiny) mass for the third one [23].

– 3 –

JHEP07(2011)122

Light neutrino best fit values

∆m221 = (7.64+0.19

−0.18 ) × 10−5 eV2

∆m231 =

{

(2.45 ± 0.09) × 10−3 eV2 NH

−(2.34+0.10−0.09 ) × 10−3 eV2 IH

sin2 θ12 = 0.316 ± 0.016

sin2 θ23 =

{

0.51 ± 0.06 NH

0.52 ± 0.06 IH

sin2 θ13 =

{

0.017+0.007−0.009 NH

0.020+0.008−0.009 IH

Table 1. The best fit values of the light neutrino parameters and their 1σ errors from [27].

where cij ≡ cos θij and sij ≡ sin θij. In addition to the Dirac-type phase δ, analogous to that

of the quark sector, there are two physical phases φi if neutrinos are Majorana particles.

The measurement of these parameters is by now restricted to oscillation experiments which

are only sensitive to mass-squared splittings (∆m2ij ≡ m2

i − m2j). Moreover, oscillations in

vacuum cannot determine the sign of the splittings. As a consequence, an uncertainty in

the ordering of the masses remains; the two possibilities are:

m1 < m2 < m3 , (2.2)

m3 < m1 < m2 . (2.3)

The first option is the so-called normal hierarchy spectrum while the second one is the

inverted hierarchy scheme; in this form they correspond to the two possible choices of the

sign of ∆m231 ≡ ∆m2

atm, which is still undetermined, while ∆m221 ≡ ∆m2

sol is known to be

positive. Within this minimal context, two mixing angles and two mass-squared splittings

are relatively well determined from oscillation experiments (see table 1), there is a slight

hint of θ13 > 0 and nothing is known about the phases.

Regarding the absolute neutrino mass scale, it is constrained by laboratory experiments

searching for its kinematic effects in Tritium β-decay, which are sensitive to the so-called

effective electron neutrino mass,

m2νe

≡∑

i

m2i |Uei|2. (2.4)

The present upper limit is mνe < 2.2 eV at 95% confidence level (CL) [28, 29], while a

new experimental project, KATRIN [30], is underway, with an estimated sensitivity limit

mνe ∼ 0.2 eV. However, cosmological observations provide the tightest constraints on

the absolute scale of neutrino masses, via their contribution to the energy density of the

Universe and the growth of structure. In general these bounds depend on the assumptions

made about the expansion history as well as on the cosmological data included in the

– 4 –

JHEP07(2011)122

analysis [31]. Combining CMB and large scale structure data quite robust bounds have been

obtained:∑

i mi < 0.4 eV at 95% CL within the ΛCDM model [32] and∑

i mi < 1.5 eV at

95% CL when allowing for several departures from ΛCDM [33].

Finally, if neutrinos are Majorana particles complementary information on neutrino

masses can be obtained from 0ν2β decay. The contribution of the known light neutrinos

to the 0ν2β decay amplitude is proportional to the effective Majorana mass of νe, mee =

|∑i miU2ei|, which depends not only on the masses and mixing angles of the UPMNS matrix

but also on the phases. The present bound from the Heidelberg-Moscow group is mee <

0.34 eV at 90% CL [34], but future experiments can reach sensitivities of up to mee ∼0.01 eV [35].

We now briefly review the current status of searches for new sequential generations.

Direct production of the 4th generation quarks t′ and b′, assuming t′ → Wq and b′ →Wt has been searched in CDF, leading to the lower mass bounds mt′ > 335 GeV and

mb′ > 385 GeV [36, 37]. Limits on new generation leptons, from LEP II, are weaker:

mℓ′ > 100.8 GeV and mν′ > 80.5 (90.3) GeV for pure Majorana (Dirac) particles, assuming

that the 4th generation leptons are unstable, i.e., their mixing with the known leptons

is large enough so that they decay inside the detector [17]. When neutrinos have both

Dirac and Majorana masses, their coupling to the Z boson may be reduced by the neutrino

mixing angle and the bound on the lightest neutrino mass may be relaxed to 63 GeV [38].

While the bound on a charged lepton stable on collider lifetimes is still about 100 GeV,

in the case of stable neutrinos the only limit comes from the LEP I measurement of the

invisible Z width, mν′ > 39.5 (45) GeV for pure Majorana (Dirac) particles [17].

Even if new generation fermions are very heavy and cannot de directly produced, they

affect electroweak observables through radiative corrections. Recent works have shown that

a fourth generation is consistent with electroweak precision observables [20, 39, 40], pro-

vided there is a heavy Higgs and the mass splittings of the new SU(2) doublets satisfy [40]2

|mt′ − mb′ | < 80GeV, (2.5)

|mℓ′ − mν′ | < 140GeV. (2.6)

Notice, however, that a long-lived fourth generation can reopen a large portion of the

parameter space [41].

In addition to these phenomenological bounds one can place some upper limits by

using perturbative unitarity, triviality and by imposing the stability of the Higgs potential

at one loop. Typically one obtains limits of the order of the TeV [42] for degenerate lepton

doublets and about 600 GeV for degenerate quark doublets.

A very striking effect of new generations is the enhancement of the Higgs-gluon-gluon

vertex which arises from a triangle diagram with all quarks running in the loop. This

vertex is enhanced approximately by a factor 3 (5) in the presence of a heavy fourth (fifth)

generation [39, 43]. Therefore, the Higgs production cross section through gluon fusion at

the Tevatron and the LHC is enhanced by a factor of 9 (25) in the presence of a fourth (fifth)

2The allowed quark mass splittings depend on the Higgs mass, according to the approximate formula

mt′ − mb′ ≃

`

1 + 1

5log( mH

115 GeV)´

× 50 GeV from ref. [39].

– 5 –

JHEP07(2011)122

νi

νj

eα

W

W

eβν4, ν 4

Figure 1. Two-loop diagram contributing to neutrino masses in the four-generation model.

generation. Thus, a combined analysis from CDF and D0 for four generations has excluded

a SM-like Higgs boson with mass between 131 GeV and 204 GeV at 95% CL [44], while

LHC data already excludes 144GeV < mH < 207GeV at 95% CL [45]. From these results,

we estimate roughly that mH > 300 GeV in the case of five generations. However, these

limits may be softened if the fourth generation neutrinos are long-lived and the branching

ratio of the decay channel H → ν4ν4 is significant [46].

Putting all together, a general analysis seems to suggest that at most only two extra

generations are allowed [19] unless new additional physics is invoked. If extra generations

exist, the Higgs should be heavy. Extra generation quarks should also be quite heavy and

be almost degenerate within a generation. The constraints on new generation leptons are

milder; charged lepton and Dirac neutrino masses should be in the range 100–1000 GeV

and, as we will see in section 4.3, this range will increase if neutrinos have both Dirac and

Majorana mass terms.

3 Four generations

If we add one right-handed neutrino νR to the SM with three generations and we do not

impose lepton number conservation, so that there is a Majorana mass term for the right-

handed neutrino, a particular linear combination of νe, νµ, ντ , call it ν ′3, will couple to νR

and get a Majorana mass at tree level. The other two linear combinations are massless at

tree level but, since lepton number is broken, no symmetry protects them from acquiring

a Majorana mass at the quantum level. In fact, they obtain a mass at two loops by the

exchange of two W bosons (same diagram as in figure 1, but with ν3, ν3 running in the

loop). This leads to two extremely small neutrino masses, as desired, but there is a huge

hierarchy between the tree-level mass, for ν3, and the two-loop-level masses, for ν1 and ν2,

therefore this possibility cannot accommodate the observed neutrino masses.3

Analogously, we can extend the SM by adding a complete fourth generation and one

right handed neutrino νR with a Majorana mass term [22, 26, 48, 49]. We denote the new

3See however [47] for a model with three generations, one right-handed neutrino singlet and two Higgs

doublets which can accommodate neutrino masses and mixings.

– 6 –

JHEP07(2011)122

charged lepton E and the new neutrino νE . The relevant part of the Lagrangian is

LY = −ℓYeeRφ − ℓYννRφ − 1

2νc

RmRνR + H.c. , (3.1)

where ℓ represents the left-handed lepton SU(2) doublets, eR the right-handed charged

leptons, νR the right-handed singlet and flavour indices are omitted. In generation space

ℓ and eR are organized as column vectors with four components. Thus, Ye is a general,

4 × 4 matrix, Yν is a general four-component column vector whose elements we denote by

yα with α = e, µ, τ,E, and mR is a Majorana mass term. The standard kinetic terms, not

shown in eq. (3.1), are invariant under general unitary transformations ℓ → Vℓℓ, eR → VeeR

and νR → eiανR. One can use those transformations, Vℓ and Ve, to choose Ye diagonal and

positive and also mR can be taken positive by absorbing its phase in νR. Yν is in general

arbitrary; however, there is still a rephasing invariance in ℓ and eR that will allow us to

remove all phases in Yν .

After spontaneous symmetry breaking (SSB) the mass matrix for the neutral leptons is

a 5×5 Majorana symmetric matrix which has the standard see-saw structure with only one

right-handed neutrino Majorana mass term. Therefore, it leads to two massive Majorana

and three massless Weyl neutrinos. From the Lagrangian it is clear that only the linear

combination of left-handed neutrinos ν ′4 ∝ yeνe + yµνµ + yτντ + yEνE will pair up with

νR to acquire a Dirac mass term. Thus, it is convenient to pass from the flavour basis

(νe, νµ, ντ , νE) to a new one ν ′1, ν

′2, ν

′3, ν

′4 where the first three states will be massless at

tree level and only ν ′4 will mix with νR. If V is the orthogonal matrix that passes from

one basis to the other we will have να =∑

i Vαiν′i (i = 1, · · · , 4, α = e, µ, τ,E) with

Vα4 ≡ Nα = yα/√

∑

β y2β. Since ν ′

1, ν′2, ν

′3 are massless, we are free to choose them in

any combination of νe, νµ, ντ , νE as long as they are orthogonal to ν ′4, i.e.,

∑

α VαiNα = 0

for i = 1, 2, 3. The orthogonality of V almost fixes all its elements in terms of Nα, but

still leaves us some freedom to set three of them to zero. Following [22, 48] we choose

Vτ1 = VE1 = VE2 = 0 for convenience.

After this change of basis, we are left with a non-trivial 2 × 2 mass matrix for ν ′4 and

νR which can easily be diagonalized and leads to two Majorana neutrinos

ν4 = i cos θ(−ν ′4 + ν ′c

4 ) + i sin θ(νR − νcR) ,

ν4 = − sin θ(ν ′4 + ν ′c

4 ) + cos θ(νR + νcR) ,

m4,4 =1

2

(√

mR2 + 4m2

D ∓ mR

)

, (3.2)

where mD = v√

∑

i y2i , with v = 〈φ(0)〉, and tan2 θ = m4/m4. The factor i and the

relative signs in ν4 are necessary to keep the mass terms positive and preserve the canonical

Majorana condition ν4 = νc4. If mR ≪ mD, we have m4 ≈ m4, tan θ ≈ 1, and we say we

are in the pseudo-Dirac limit while when mR ≫ mD, m4 ≈ m2D/mR and m4 ≈ mR,

tan θ ≈ mD/mR and we say we are in the see-saw limit.

Since lepton number is broken by the νR Majorana mass term, there is no symmetry

which prevents the tree-level massless neutrinos from gaining Majorana masses at higher

– 7 –

JHEP07(2011)122

order. In fact, Majorana masses for the light neutrinos, ν ′1, ν

′2, ν

′3, are generated at two

loops by the diagram of figure 1, and are given by

Mij = − g4

m4W

mRm2D

∑

α

VαiVα4m2α

∑

β

VβjVβ4m2βIαβ , (3.3)

where the sums run over the charged leptons α, β = e, µ, τ,E while i, j = 1, 2, 3, and

Iαβ = J(m4,m4,mα,mβ, 0) − 3

4J(m4,m4,mα,mβ,mW ) , (3.4)

with J(m4,m4,mα,mβ,mW ) the two-loop integral defined and computed in appendix A.

When mR = 0, Mij = 0, as it should, because in that case lepton number is conserved.

Also when mD = 0 we obtain Mij = 0, since then the right-handed neutrino decouples

completely and lepton number is again conserved.

To see more clearly the structure of this mass matrix we can take, for the moment,

the limit me = mµ = mτ = 0; then, since we have chosen Vτ1 = VE1 = VE2 = 0, the only

non-vanishing element in Mij is M33 and it is proportional to V 2E3N

2Em4

EIEE. Keeping

all the masses one can easily show that the eigenvalues of the light neutrino mass matrix

are proportional to m4µ, m4

τ , m4E which gives a huge hierarchy between neutrino masses.

Moreover, for mE ≫ m4,4 ≫ mW , the loop integrals in eq. (3.4) can be well approximated

by (see appendix A):

IEE ≈ −1

(4π)42m2E

lnmE

m4(3.5)

and

Iµµ ≈ Iττ ≈ −1

(4π)42m24

lnm4

m4, (3.6)

leading to only two light neutrino masses, since the mass matrix in eq. (3.3) has rank 2 if

the three light charged lepton masses are neglected in Iαβ. The third light neutrino mass is

generated when at least mτ is taken into account in the loop integral, leading to a further

suppression. Within the above approximation, the following ratio of ν2 and ν3 masses is

obtained [50]:

m2

m3.

1

4N2E

(

mτ

mE

)2(mτ

m4

)2

.10−7

N2E

, (3.7)

where we have taken ln(m4/m4) ≈ ln(mE/m4) ≈ 1 and in the last step we used that

mE ,m4 & 100 GeV. To overcome this huge hierarchy one would need very small values of

NE which would imply that the heavy neutrinos are not mainly νE but some combination of

the three known neutrinos νe, νµ, ντ ; but this is not possible since it would yield observable

effects in a variety of processes, like π → µν, π → eν, τ → eνν, τ → µνν. This requires

that ye,µ,τ . 10−2yE [51, 52] and then NE ≈ 1.

Therefore, although the idea is very attractive, the simplest version is unable to ac-

commodate the observed spectrum of neutrino masses and mixings. However, notice that

whenever a new generation and a right-handed neutrino with Majorana mass at (or below)

the TeV scale are added to the SM, the two-loop contribution to neutrino masses is always

present and provides an important constraint for this kind of SM extensions.

– 8 –

JHEP07(2011)122

In the following we modify the original idea by adding one additional generation and

one additional fermion singlet. We will see that this minimal modification is able to ac-

commodate all current data.

4 Five generation working example

4.1 The five generations model

We add two generations to the SM and two right-handed neutrinos. We denote the two

charged leptons by E and F and the two right-handed singlets by ν4R and ν5R. The

Lagrangian is exactly the same we used for four generations (3.1) but now ℓ and e are

organized as five-component column vectors while νR is a two-component column vector

containing ν4R and ν5R. Thus, Ye is a general, 5 × 5 matrix, Yν is a general 5 × 2 matrix

and mR is now a general symmetric 2 × 2 matrix. The kinetic terms are invariant under

general unitary transformations ℓ → Vℓℓ, eR → VeeR and ν → Vνν, which can be used to

choose Ye diagonal and positive and mR diagonal with positive elements m4R and m5R.

After this choice, there is still some rephasing invariance ℓi → eiαiℓi, eiR → eiαieiR broken

only by Yν , which can be used to remove five phases in Yν . Therefore

Yν =(

y, y′)

, (4.1)

where y and y′ are five-component column vectors with components yα and y′α respectively

(α = e, µ, τ,E, F ), one of which can be taken real while the other, in general, will contain

phases. The model, contrary to the four-generation case, has additional sources of CP

violation in the leptonic sector. However, since at the moment we are not interested in CP

violation, for simplicity we will take all yα and y′α real.

Much as in the four-generation case, the linear combination ν ′4 ∝ ∑

α yανα only couples

to ν4R and the combination ν ′5 ∝ ∑

α y′ανα only couples to ν5R. Therefore, the tree-level

spectrum will contain three massless neutrinos (the linear combinations orthogonal to ν ′4

and ν ′5) and four heavy Majorana neutrinos. Unfortunately, since in the general case ν ′

4

and ν ′5 may not be orthogonal to each other, the diagonalization becomes much more

cumbersome than in the four-generation case. Since we just want to provide a working

example, we choose ν ′4 and ν ′

5 orthogonal to each other, i.e.,∑

α yαy′α = 0. This simplifies

enormously the analysis of the model and allows us to adopt a diagonalization procedure

analogous to the one followed in the four-generation case.

We change from the flavour fields νe, νµ, ντ , νE , νF to a new basis ν ′1, ν

′2, ν

′3, ν

′4, ν

′5 where

ν ′1, ν

′2, ν

′3 are massless at tree level, so we are free to choose them in any combination of

the flavour states as long as they are orthogonal to ν ′4 and ν ′

5. Thus, if V is the orthogonal

matrix that passes from one basis to the other να =∑

i Vαiν′i (i = 1, · · · , 5, α = e, µ, τ,E, F )

we have Vα4 = Nα = yα/√

∑

β y2β, Vα5 = N ′

α = y′α/√

∑

β y′2β , and∑

β NβN ′β = 0. The

rest of the elements in Vαi can be found by using the orthogonality of V , which gives us 12

equations (9 orthogonality and 3 normalization conditions, because Nα and N ′α are already

normalized and orthogonal), therefore we still can choose at will three elements of Vαi; for

– 9 –

JHEP07(2011)122

instance we could choose VF1 = VF2 = VE1 = 0. In this case

V =

Ve1 Ve2 Ve3 Ne N ′e

Vµ1 Vµ2 Vµ3 Nµ N ′µ

Vτ1 Vτ2 Vτ3 Nτ N ′τ

0 VE2 VE3 NE N ′E

0 0 VF3 NF N ′F

. (4.2)

Moreover, since∑

α NαN ′α = 0, the 4×4 mass matrix of ν ′

4, ν4R, ν ′5 and ν5R is block-diagonal

and can be separated in two 2 × 2 matrices (for ν ′4 and ν4R and ν ′

5 and ν5R respectively)

with the same form found in the four-generation case. Its diagonalization leads to four

Majorana massive fields:

νa = i cos θa(−ν ′a + ν ′c

a ) + i sin θa(νaR − νcaR) ,

νa = − sin θa(ν′a + ν ′c

a ) + cos θa(νaR + νcaR) ,

ma,a =1

2

(√

m2aR + 4m2

aD ∓ maR

)

, (4.3)

with a = 4, 5, tan2 θa = ma/ma, m4D = v√

∑

α y2α and m5D = v

√∑

α y′2α .

4.2 Two-loop neutrino masses

As in the case of four generations, the diagrams of figure 1 (now with the four massive

neutrinos running in the loop) will generate a non-vanishing mass matrix for the three

neutrinos ν ′1, ν

′2, ν

′3 given by

Mij = − g4

m4W

∑

a=4,5

maRm2aD

∑

α

VαiVαam2α

∑

β

VβjVβam2βI

(a)αβ , (4.4)

with I(a)αβ given by (3.4) with a labeling the contribution of the 4th and 5th generations.

To analyze this mass matrix first we will impose several phenomenological constraints:

a) The model should be compatible with the observed universality of fermion couplings

and have small rates of lepton flavour violation in the charged sector. This requires

ye, yµ, yτ , y′e, y

′µ, y′τ ≪ yE, yF , y′E, y′F .

b) The model should fit the observed pattern of masses and mixings. A good starting

point would be to have expressions able to reproduce the tribimaximal (TBM) mixing

structure.

The constraint a) together with the orthogonality condition implies that yEy′E +yF y′F ≈ 0,

which can be satisfied, for instance, if yF = y′E = 0, that is, νE only couples to ν4R and νF

only couples to ν5R. Then, one can define yα = yE(ǫe, ǫµ, ǫτ , 1, 0), y′α = y′F (ǫ′e, ǫ′µ, ǫ′τ , 0, 1),

where ǫi and ǫ′i are at least O(10−2) in order to satisfy universality constraints4 (see sec-

tion 4.3.2 for more details). Thus, to order ǫ, Nα ≈ (ǫe, ǫµ, ǫτ , 1, 0), N ′α ≈ (ǫ′e, ǫ

′µ, ǫ′τ , 0, 1),

4This pattern of couplings can easily be enforced by using a discrete symmetry which is subsequently

broken at order ǫ.

– 10 –

JHEP07(2011)122

and since for i 6= 4, 5∑

α VαiNα =∑

α VαiN′α = 0, all the entries Vαi with α = e, µ, τ ,

i = 1, 2, 3 can be order one. Now if we choose VF1 = VF2 = VE1 = 0 one can see that

VE2, VE3 are O(ǫ) while VF3 is O(ǫ′).

A further simplification occurs if we assume that VE3 = 0, since in that case E only

couples to ν ′2 and and F only couples to ν ′

3. Then, in the limit me = mµ = mτ = 0 the

neutrino mass matrix Mij in eq. (4.4) is already diagonal and we can easily estimate the

size of the two larger eigenvalues by neglecting the masses of the known charged leptons

in front of mE and mF . We find5

M22 ∼ ǫ2m4Rg4m2

4Dm2E

m4W (4π)42

lnmE

m4

, M33 ∼ ǫ′2m5Rg4m2

5Dm2F

m4W (4π)42

lnmF

m5

. (4.5)

Taking mF ∼ m5D ∼ mW /g and ǫ′ ∼ 10−2 we find M33 ∼ 2 × 10−9m5R, therefore, to

obtain M33 ∼ 0.05 eV we need m5R ∼ 20 MeV (or ǫ′ . 10−3 for m5R ∼ 1GeV). Since m5R

and m4R control the splitting between the two heavy Majorana neutrinos we are naturally

in the pseudo-Dirac regime unless the ǫ’s are below 10−4. We also see that the higher

m4D(5D) or mE(F ), the lower the ǫ (ǫ′) that is needed for a given m4R(5R). On the other

hand, it is clear that the required hierarchy between M33 and M22 can be easily achieved

both in the normal and the inverted hierarchy cases, while the degenerate case cannot

be fitted within this scheme since the third neutrino mass is proportional to m4τ . After

discussing the phenomenology of the model with more detail in section 4.3, we present the

allowed regions of the parameter space in figure 4.

Now let us turn to constraint b), that is, the light neutrino mixings. With our simpli-

fying choices the diagonal entries of the light neutrino mass matrix are proportional to m4τ ,

m4E , m4

F , whereas the off-diagonal ones are proportional to m2τm

2E and m2

τm2F . Therefore

the neutrino states ν ′1, ν

′2, ν

′3 are very close to being the true mass eigenstates and the first

3 × 3 elements of V , Vαi, with α = e, µ, τ, i = 1, 2, 3 give us directly the PMNS mixing

matrix (up to permutations). Then, by using the orthogonality conditions it is easy to

find the structure of Yukawas that reproduce a given pattern for the PMNS matrix. Let

us study separately the two phenomenologically viable cases, normal hierarchy (NH) and

inverted hierarchy (IH).

4.2.1 Normal hierarchy

In the normal hierarchy case (m1 < m2 < m3), the experimental data tell us that

m3 ≈√

| △ m231| ≈ 0.05 eV, m2 ≈

√

△m221 ≈ 0.01 eV and allow for m1 ≪ m2. The struc-

tures we have found (by choosing VF1 = VF2 = VE1 = VE3 = 0) automatically fall in this

scheme, since (for mE,F ≫ m4,4,5,5 ≫ mW ) we obtain (m1,m2,m3) ∝ (m4τ/m

24,m2

E ,m2F ).

Is there any choice of the Yukawa couplings yα and y′α that leads naturally to some

phenomenologically successful structure, for instance TBM? If we impose TBM in Vαi

(α = e, µ, τ , i = 1, 2, 3), given the structure of Nα and N ′α, the orthogonality of V (at

order ǫ2) immediately tells us that ǫe = ǫµ = −ǫτ ≡ ǫ, ǫ′e = 0, ǫ′µ = ǫ′τ ≡ ǫ′, and finally

5Note that the position of the eigenvalues in Mij depends on the position of the zeros in Vαi. The choice

we made is very convenient to reproduce the normal hierarchy spectrum.

– 11 –

JHEP07(2011)122

VE2 = −ǫ√

3, VF3 = −ǫ′√

2. Therefore, a successful choice of the Yukawas will be

yα = yE(ǫ, ǫ,−ǫ, 1, 0) ,

y′α = y′F (0, ǫ′, ǫ′, 0, 1) , (4.6)

which, keeping only terms up to order ǫ2, leads to

V ≈

√

2

3

1√

3−

√3

2ǫ2 0 ǫ 0

−1√

6

1√

3−

√3

2ǫ2

1√

2−

1√

2ǫ′2 ǫ ǫ′

1√

6−

1√

3+

√3

2ǫ2

1√

2−

1√

2ǫ′2 −ǫ ǫ′

0 −ǫ√

3 0 1 −3

2ǫ2 0

0 0 −ǫ′√

2 0 1 − ǫ′2

+ O(ǫ3) . (4.7)

Assuming that mE,F ≫ m4,4,5,5 ≫ mW , we find:

m2 = − 3g4

m4W

ǫ2m24Dm4Rm4

EIEE ≈ 3g4

2(4π)4m4W

ǫ2m24Dm4Rm2

E lnmE

m4

, (4.8)

m3 = − 2g4

m4W

ǫ′2m25Dm5Rm4

F IFF ≈ g4

(4π)4m4W

ǫ′2m25Dm5Rm2

F lnmF

m5

, (4.9)

and the required ratio m3/m2 ≈ 5 can be easily accommodated, for instance if the fifth

generation is heavier than the fourth one or ǫ′ > ǫ.

4.2.2 Inverted hierarchy

In the inverted hierarchy case (m3 < m1 . m2), we have m2 ≈ m1 ≈√

| △ m231| ≈ 0.05 eV

and m3 ≪ m1 is allowed. Therefore now we need (m1,m2,m3) ∝ (m2E ,m2

F ,m4τ/m

24), which

is just a cyclic permutation of the three eigenvalues. This ordering cannot be obtained

directly with our previous choice for the zeroes in Vαi, so now it is convenient to choose

VF1 = VF3 = VE3 = VE2 = 0 instead. Following the same procedure as above, we find that

yα = yE(−2ǫ, ǫ,−ǫ, 1, 0) ,

y′α = y′F (ǫ′, ǫ′,−ǫ′, 0, 1) (4.10)

will reproduce the desired TBM pattern, leading to

V ≈

√

2

3−

√6ǫ2

1√

3−

√3

2ǫ′2 0 −2ǫ ǫ′

−1√

6+

√

32 ǫ2

1√

3−

√3

2ǫ′2

1√

2ǫ ǫ′

1√

6−

√

32 ǫ2 −

1√

3+

√3

2ǫ′2

1√

2−ǫ −ǫ′

ǫ√

6 0 0 1 − 3 ǫ2 0

0 −ǫ′√

3 0 0 1 −3

2ǫ′2

+ O(ǫ3) . (4.11)

– 12 –

JHEP07(2011)122

Assuming that mE,F ≫ m4,4,5,5 ≫ mW , we get:

m1 ≈ 3g4

(4π)4m4W

ǫ2m24Dm4Rm2

E lnmE

m4, (4.12)

m2 ≈ 3g4

2(4π)4m4W

ǫ′2m25Dm5Rm2

F lnmF

m5, (4.13)

while the ratio of masses between the heaviest neutrinos, m1/m2 ≈ 1 can be obtained by

choosing the different parameters in their natural range.

Thus, the model accommodates the light neutrino masses and mixings. In the next

section we will analyse current phenomenological bounds on the mixings between the new

generations and the first three, ǫ, ǫ′.

4.3 The parameters of the model

We have seen above that neutrino masses are proportional to ǫ2m4R (or ǫ′2m5R) and a

product of masses, m24Dm2

E (or m25Dm2

F ), which come from the Higgs mechanism and are

proportional to Yukawa couplings. As discussed in section 2 the values of these masses can-

not vary too much; perturbative unitarity requires they are smaller than about 1TeV [42]

while lower limits for charged leptons masses from colliders are about 100 GeV. Lower

limits for neutral fermions are a bit less uncertain. In the case of unstable pure Dirac

neutrinos (maR = 0, a = 4, 5) the neutrino masses are basically maD and the lower limits

are about 90 GeV, therefore, in that case, maD & 90 GeV. If neutrinos have both Dirac

and Majorana mass terms (maR 6= 0) the masses are given by eq. (4.3) and the lower limits

are6 ma ≥ ma > 63 GeV, then the upper limits on maD < 1 TeV automatically imply

maR . 16 TeV and therefore ma . 16 TeV. More generally in figure 2 we present the

allowed regions in the plane maR vs maD given the lower bound on ma > 63 GeV and the

upper limit on maD < 1000 GeV. We also plot the lines corresponding to ma = 200, 400,

600 and 800 GeV.

To be definite we will take

100GeV < m4D,m5D,mE ,mF < 1000GeV,

63GeV . ma ≤ maD ≤ ma . 16TeV,

with mama = m2aD and ma − ma = maR. In addition there are strong constrains from

the electroweak oblique parameters which in the pure Dirac case require some degeneracy

of masses, m4D ≃ mE (m5D ≃ mF ). However, these constraints depend on the complete

spectrum of the theory (masses of quarks and leptons from new generations and the Higgs

boson mass) and are less certain. In fact, contributions from the splitting of masses in the

quark sector can be compensated in part by lepton contributions with large maR [53, 54],

which, if we do take into account the constraints set by LEP II can vary from essentially

zero (Dirac case) to 16 TeV.

6Notice that in our scenario there is a lower bound on the mixing ǫ, in order to obtain the correct scale

of light neutrino masses, which implies that the heavy neutrinos would have decayed inside the detector

at LEP.

– 13 –

JHEP07(2011)122

Figure 2. Allowed region in maR–maD given the present lower limit (ma > 63GeV) on the mass

of an extra generation neutrino (a = 4, 5 refers the 4th or 5th generation).

The other parameters that enter neutrino masses are the ǫ’s, which characterize the

mixing of light neutrinos with heavy neutrinos, and the mR’s, which characterize the

amount of total lepton number breaking. The ǫ parameters will produce violations of uni-

versality and flavour lepton number conservation in low energy processes. The combination

of data from these processes will allow us to constrain both ǫ and ǫ′. On the other hand, to

obtain information on the mR’s we will use the light neutrino masses, which in the model

are Majorana particles. We will also study the contributions of the heavy neutrinos to

neutrinoless double beta decay.

4.3.1 Lepton flavour violation processes (µ → eγ and µ–e conversion)

The general expression for the branching ratio of µ → eγ produced through a virtual pair

W -neutrino is:

B(µ → eγ) =3α

2π|δν |2, (4.14)

where

δν =∑

i

UeiU∗µi H(m2

χi/m2

W) (4.15)

and H is the loop function for this process [51]

H(x) =x(2x2 + 5x − 1)

4(x − 1)3− 3x3 log(x)

2(x − 1)4,

with mχithe masses of all heavy neutrinos running in the loop and Uei and Uµi their

couplings to the electron and the muon respectively. In (4.14) we have used the unitarity

of the mixing matrix and neglected the light neutrino masses to rewrite the final result only

in terms of the heavy neutrino contributions. Then, as the mixings of the heavy neutrinos

with the light leptons are different in normal and inverted hierarchy, so are the µ → eγ

amplitudes generated; one can see just by inspection of the mixing matrices, eqs. (4.7)

– 14 –

JHEP07(2011)122

Ε = 0.025

Ε = 0.008

Current bound

Future bound

100 500200 300150 70010-14

10-13

10-12

10-11

10-10

Heavy neutrino mass m4 HGeVL

BHΜ�

eΓL

Figure 3. B(µ → eγ) against m4 for different values of ǫ in the NH case. We also display present

and future limits on B(µ → eγ) as horizontal lines.

and (4.11) that in NH only the pair ν4, ν4 couples to both the electron and the muon,

whereas in IH the four heavy neutrinos contribute to the process. The predicted branching

ratios are

NH: B(µ → eγ) =3α

2πH2

4 ǫ4, (4.16)

IH: B(µ → eγ) =3α

2π

[

H5 ǫ′2 − 2 H4 ǫ2]2

, (4.17)

where

Ha ≡ cos2 θaH(m2a/m

2W ) + sin2 θaH(m2

a/m2W ) .

Now since H(x) is a monotonically increasing function and ma ≥ ma > 63GeV we have

Ha ≥ H(m2a/m

2W ) > 0.09 which gives the less stringent constraint on ǫ and ǫ′. The

experimental bound reads B(µ → eγ) < 1.2 × 10−11, and it is translated into

NH: ǫ < 0.03 , (4.18)

IH: |ǫ′2 − 2 ǫ2| < 7 × 10−4. (4.19)

To see how these bounds depend on the masses of the heavy neutrinos we display in

figure 3 B(µ → eγ) against the mass of the heavy neutrino m4 in the NH case. For IH we

expect similar results unless there are strong cancellations. We also display as horizontal

lines present limits [17] B(µ → eγ) < 1.2 × 10−11 and near future limits [55]. From the

figure we can extract a conservative bound of the order of the one quoted above, ǫ < 0.03.

Some extra information could be extracted from τ → eγ and τ → µγ. Thus, from

B(τ → eγ) < 3.3 × 10−8 we obtain ǫ < 0.3 in the case of NH and |ǫ′2 − 2 ǫ2| < 0.08, limits

that show exactly the same dependence on ǫ and ǫ′ as the one obtained in µ → eγ, but

– 15 –

JHEP07(2011)122

which are roughly one order of magnitude worse. From B(τ → µγ) < 4.4× 10−8, although

the bounds are of the same order of magnitude as for B(τ → eγ), we obtain different

combinations of ǫ’s, |ǫ′2 − ǫ2| < 0.09 for NH and ǫ′2 + ǫ2 < 0.09 for IH.

Another very interesting process which gives information on ǫ is µ–e conversion in

nuclei. From present data [17] one obtains bounds similar to the limit obtained from

µ → eγ. However, there are plans to improve the sensitivity in µ–e conversion in 4 and

even 6 orders of magnitude [56], therefore we expect much stronger bounds in the future

coming from µ–e conversion. Strong correlations between both processes exist, as can be

seen in [51].

4.3.2 Universality bounds

New heavy generations that couple to the observed fermions can potentially lead to viola-

tions of universality in charged currents because of the “effective” lack of unitarity in the

mixings when the heavy generations cannot be produced. Data from neutrino oscillation

experiments can also be used to constrain deviations from unitarity of some of the elements

of the leptonic mixing matrix [57], however, in our scenario they lead to weaker bounds

than the ones obtained here.

There are different types of universality bounds which constrain the mixings of light

fermions with new generations:

• Lepton-hadron universality. One compares weak couplings of quarks and leptons

using muon decay and nuclear β decay, which are very well tested. In our case this

involves mixings both in the quark and lepton sectors and they are not useful to test

individually the lepton mixings we are interested in.

• Relations between muon decay, mZ , mW and the weak mixing angle sin2 θW . These

are very well-determined relations in the SM, and in our case they are modified

because the heavy neutrinos cannot be produced in ordinary muon decay. Unfortu-

nately, these relations depend strongly on the ρ parameter, which receives contribu-

tions from the Higgs and very large contributions from the heavy fermions of the new

generations. Therefore, although these type of relations could be used to set bounds

on the ǫ’s, they would depend on other unknown parameters.

• Ratios of decay widths of similar processes. The bounds obtained from this type of

processes are very robust because most of the uncertainties cancel in the ratios. We

will only consider the most precise among these ratios, which are well measured and

can be computed accurately [58–60]:

Rπ→e/π→µ ≡ Γ(π → eν)

Γ(π → µν), (4.20)

Rτ→e/τ→µ ≡ Γ(τ → eνν)

Γ(τ → µνν), (4.21)

Rτ→e/µ→e ≡ Γ(τ → eνν)

Γ(µ → eνν)= Bτ→e

τµ

ττ, (4.22)

– 16 –

JHEP07(2011)122

Observable R [17] RSM R/RSM

Rπ→e/π→µ (1.230 ± 0.004) × 10−4 (1.2352 ± 0.0001) × 10−4 0.996 ± 0.003.

Rτ→e/τ→µ 1.028 ± 0.004 1.02821 ± 0.00001 1.000 ± 0.004

Rτ→µ/µ→e (1.31 ± 0.06) × 106 (1.3086 ± 0.0006) × 106 1.001 ± 0.004

Table 2. Relevant universality tests.

Rτ→µ/µ→e ≡ Γ(τ → µνν)

Γ(µ → eνν)= Bτ→µ

τµ

ττ, (4.23)

where Bτ→f = Γ(τ → f νν)/Γ(τ → all) is the branching ratio of the tau decay to the

fermion f , and τf = 1/Γ(f → all) its lifetime. In our model there are corrections to these

ratios because νe, νµ and ντ have a small part of ν4,4 and ν5,5, which are heavy and cannot

be produced. This leads to an additional violation of universality which depends on the

mixings of νe, νµ and ντ with ν4,4 and ν5,5. For Rπ→e/π→µ, and using the Vαi in (4.7)

and (4.11), we find that

Rπ→e/π→µ

RSMπ→e/π→µ

=|Ve1|2 + |Ve2|2 + |Ve3|2|Vµ1|2 + |Vµ2|2 + |Vµ3|2

=

{

1 + ǫ′2 NH

1 − 3ǫ2 IH. (4.24)

Rτ→e/τ→µ/RSMτ→e/τ→µ tests exactly the same couplings, therefore the result is the same

as in (4.24).

Rτ→e/µ→e gives a different information because it tests τ/µ universality; however, we

find that for both NH and IH Rτ→e/µ→e/RSMτ→e/µ→e = 1, and since it is independent of the

ǫ’s, this process does not give any further information. This is a consequence of our choice

for the Yukawa couplings,7 which, up to signs, are equal for the τ and µ neutrinos.

Finally for Rτ→µ/µ→e, using our mixing matrices, we find

Rτ→µ/µ→e

RSMτ→µ/µ→e

=|Vτ1|2 + |Vτ2|2 + |Vτ3|2|Ve1|2 + |Ve2|2 + |Ve3|2

=

{

1 − ǫ′2 NH

1 + 3ǫ2 IH, (4.25)

which gives exactly the inverse combinations of those obtained from e/µ universality tests.

Therefore, if we combine the three results, Rπ→e/π→µ/RSMπ→e/π→µ,Rτ→e/τ→µ/RSM

τ→e/τ→µ

and (Rτ→µ/µ→e/RSMτ→µ/µ→e)

−1 and use the data collected in table 2, we obtain

0.998 ± 0.002 =

{

1 + ǫ′2 NH

1 − 3ǫ2 IH, (4.26)

which translates into the following upper 90% CL limits on ǫ′ and ǫ

NH: ǫ′ < 0.04 , (4.27)

IH: ǫ < 0.04 . (4.28)

7Which, in turn, is a consequence of the TBM structure we wanted to reproduce.

– 17 –

JHEP07(2011)122

Notice that although in the IH case we have more sensitivity than in the NH case because

of the factor of 3 we finally obtain similar limits in the two cases. This is because in the

NH case the deviation from 1 obtained in the model is always positive while the present

measured value is slightly smaller than 1 (in both cases we used the Feldman & Cousins

prescription [61] to set 90% CL limits).

Now we can use all data from LFV and universality and conclude that in the NH case

we have ǫ < 0.03, basically from µ → eγ, and ǫ′ < 0.04, basically from universality tests.

In the IH case we obtain that, except in a narrow band around ǫ′2 ≃ 2ǫ2, ǫ . 0.02 and

ǫ′ . 0.03 basically from µ → eγ; if ǫ′2 ≃ 2ǫ2 there is a cancellation in µ → eγ but still one

can combine these data with the universality limits to obtain ǫ . 0.04 and ǫ′ . 0.06.

4.3.3 Neutrinoless double beta decay (0ν2β)

As commented above m4R,m5R are the relevant parameters which encapsulate the non-

conservation of total lepton number and they control, together with the ǫ’s, the neutrino

masses. Therefore, it would be useful to have additional independent information on these

parameters. The most promising experiments to test the non-conservation of total lep-

ton number are neutrinoless double beta decay experiments. The standard contribution,

produced by light neutrinos, to 0ν2β has largely been studied (for a recent review see for

instance [35]) and, given the expected future sensitivity, mee = 0.01 eV, it will be very

difficult to see it unless the neutrino spectrum is inverted or degenerate. However, if heavy

neutrinos from new families are Majorana particles, they lead to tree-level effects in neu-

trinoless double beta decay [62], while, in our scenario, light neutrino masses are generated

at two loops; thus, in principle, it is possible that these new contributions dominate over

the standard ones.

The contribution of new generation heavy neutrinos (with mass larger than about the

proton mass, mp) to the rate of neutrinoless double beta decay can be written in terms of

an effective mass,

〈MN 〉−1 =∑

a

U2eaM

−1a , (4.29)

where Uea is the coupling of the electron to the left-handed component of the heavy neu-

trino a. The non-observation of neutrinoless double beta decay implies that [63]

〈MN 〉 > 108 GeV. (4.30)

In the case of NH the electron only couples to ν4 and ν4 (see eqs. (4.7) and (4.3)), thus

〈MN 〉−1 = 2ǫ2

(

cos2 θ

m4− sin2 θ

m4

)

= ǫ2 m4R

m24D

, (4.31)

and using (4.30), we get:

m4Rǫ2/m24D < 10−8 GeV−1. (4.32)

This is the same combination of the relevant parameters (m4Rǫ2) that appears in (4.8) for

m2, therefore we can use neutrino data to set bounds on the heavy neutrino contribution

– 18 –

JHEP07(2011)122

m4 D Hm5 D) = 100 GeV

Current bound

Future boundmE HmF) = 100 GeV

m4 D Hm5 D) = 1000 GeVmE HmF) = 1000 GeV

10-6 10-4 0.01 1 100 10410-6

10-5

10-4

0.001

0.01

Majorana mass m4 R Hm5 RL HGeVL

Mix

ingΕHΕ

'L

Figure 4. Parameter space that predicts the right scale for heavy and light neutrinos (blue region

between the curves). As a comparison we also present the current bound from µ → eγ and future

limits from µ–e conversion experiments.

to neutrinoless double beta decay written in terms of the effective mass. We obtain

〈MN 〉 =

3g4m44Dm2

E lnmE

m4

2m2(4π)4m4W

& 2 × 1011 GeV, (4.33)

where we have used m2 ∼ 0.01 eV, typical values for m4D ∼ mE ∼ 100 GeV and

ln(mE/m4) ∼ 1. This is far from present, eq. (4.30), and future sensitivities.

In the case of IH, the effective mass is given by

〈MN 〉−1 = 4ǫ2m4R/m24D + ǫ′2m5R/m2

5D , (4.34)

leading also to unobservable effects in 0ν2β decay.

To summarize all phenomenological constraints on the model, in figure 4 we show

in blue the allowed region in the ǫ–m4R plane, which leads to M33 ∼ 0.05 eV varying

the charged lepton masses mE (mF ) and the Dirac neutrino masses m4D (m5D) between

100 GeV–1 TeV, and imposing the LEP bound on the physical neutrino mass, m4 > 63 GeV.

We also plot the present bounds on the mixings ǫ (ǫ′) from µ → eγ and future limits from

µ–e conversion if expectations are attained.

5 Collider signatures

As we mentioned before, the LHC offers a unique opportunity to discover (or exclude)

new sequential generations of quarks and leptons. For instance, with 1 fb−1 at 7TeV, the

exclusion bound on b′ would reach 500 GeV via b′ → Wt decay channel, close to the partial

wave unitarity bound. Even if the t′ and b′ are too heavy to be seen directly, their effects

may be manifest at LHC, since they induce a large gg → ZZ signal [64]. See also [41] for

– 19 –

JHEP07(2011)122

prospects of detecting very long-lived fourth generation quarks, i.e., in the case of extremely

small mixings with the lighter three generations.

Regarding the lepton sector, the standard searches for a fourth generation have to

be restricted to the parameter space which leads to the correct light neutrino mass scale,

depicted in figure 4. The expected signatures depend on the nature (Dirac or Majorana)

of the neutrinos, which are generally assumed to be the lightest states.

Most theoretical analysis of fourth generation Majorana neutrino at hadron colliders

have focused on the process qq′ → W± → ν4ℓ±, where the fourth generation neutrino is

produced in association with a light charged lepton [65–68]. Subsequently, if neutrinos are

Majorana, they will decay through ν4 → W∓ℓ±, leading to the low-background like-sign di-

lepton signature in half the events. However in our model the cross section for this process

is suppressed both by the mixing of the extra generations with the first three and by the

small Majorana masses m4R,5R, much as in the neutrinoless double beta decay discussed

above, so it will not be observable at LHC for the parameter range that reproduce the

correct scale of light neutrino masses.

Alternatively, the lighter neutrinos can be pair-produced via an s-channel Z boson,

qq → Z → νIνJ (I, J = 4, 4) [69]. Although the W production has a higher cross section

than the Z at hadron colliders, and the mass reach is enhanced when only one heavy

particle is produced, if the mixing angle between the extra generations and the light ones

is less than about 10−6 the neutrino production rates in the W channel are so suppressed

that they are unobservable [65]. However the rate of heavy neutrino pair-production via

a Z boson is independent of this mixing, becoming the dominant production mechanism

in the small mixing regime. Moreover, if the mass difference between ν4 and ν4 is at least

1 GeV, and the mixing so small that the decay ν4 → ν4Z always dominates, the above

processes also lead to like-sign leptons in half of the events.8 See ref. [69] for a detailed

study of the Tevatron dataset potential to exclude (or discover) fourth generation neutrinos

with both, Dirac and Majorana masses, up to 150–175 GeV, depending on the mixing. For

the LHC, only the pure Majorana case has been studied in ref. [70]. According to them,

the LHC at√

s = 10 TeV with 5 fb−1 could expect to set a 95% CL mass lower limit of

mN > 300 GeV or report 3σ evidence for the ν4 if mν4< 225 GeV. We expect a similar

sensitivity in our model, in the region m4 − m4 > 1 GeV and small mixing (ǫ, ǫ′ . 10−4)

i.e., somehow complementary to the one probed in LFV processes. See also [71] for an

evaluation of the LHC discovery potential for both Majorana and Dirac type fourth family

neutrinos in the process pp → Z/H → ν4ν4 → WµWµ.

Searches for fourth generation charged leptons at the LHC have been studied in [72],

also in a general framework with Dirac and Majorana neutrino masses, and assuming

that the neutrino ν4 is the lightest fourth generation lepton. For charged leptons with

masses under about 400 GeV, the dominant production channel is charged lepton - neutrino,

through the process qq′ → W± → ν4E±. The neutrino ν4 can only decay to ν4 → Wℓ,

and being Majorana it can decay equally to W−ℓ+ and W+ℓ−. Therefore when a pair of

8In the exact Dirac limit, ν4 must decay to Wℓ and the different contributions to same sign di-lepton

production cancel, since the Dirac neutrino conserves lepton number. However, as far as ν4 always decays

to ν4Z there is no interference amplitude, and same-sign di-lepton decays are unsuppressed.

– 20 –

JHEP07(2011)122

fourth generation leptons are produced, we expect the decay products to contain like-sign

di-leptons half of the time. The sensitivity study for this process in events with two like-sign

charged leptons and at least two associated jets shows that with√

s = 7 TeV and 1 fb−1

of data, the LHC can exclude fourth generation charged lepton masses up to 250 GeV. It

would be interesting to study the parameter space in our model that would lead to this

type of signals.

In the above searches, it was assumed that the lightest neutrino decays promptly. How-

ever, if the mixing of the lightest fourth generation neutrino with the first three generation

leptons is ǫ . 10−7 its proper lifetime will be τ4 & 10−10s. The decay length at the LHC

is given by d = βcγτ4 ∼ 3 cm(τ4/10−10s)βγ, thus for τ4 & 10−10s the fourth neutrino

will either show displaced vertices in its decay or decay outside the detector, if d & O(m),

which is a typical detector size. In our scenario, such a tiny mixing is only compatible

with large Majorana masses, mR ∼ 1 TeV (see figure 4), far from the pseudo-Dirac case.

Searches for Majorana neutrinos stable on collider times have been discussed in [73], where

it is proposed to use a quadri-lepton signal that follows from the pair production and decay

of heavy neutrinos pp → Z → ν4ν4 → ν4ν4ZZ, when both Z’s decay leptonically. The final

state is thus 4ℓ plus missing energy. For 30 fb−1 of LHC data at 13 TeV, ν4 masses can be

tested in the range 100 to 180 GeV, and ν4 masses from 150 to 250 GeV.

Finally, if the lightest fourth generation lepton is the charged one, there is a striking

signal which to our knowledge has not been studied in the literature: lepton number vio-

lating like-sign fourth generation lepton pair-production, through qq′ → W± → E±ν4,4 →E±E±W∓ or via W fusion, qq → W±W±q′q′ → E±E±jj. These processes are not sup-

pressed by the small mixing with the first three generations, so in principle they could

be observable in our scenario. Depending on the charged lepton lifetime, they will decay

promptly to same-sign light di-leptons, show displaced vertices or leave an anomalous track

of large ionization and/or low velocity. A detailed phenomenological study would be very

interesting, but it is beyond the scope of this work.

6 Summary and conclusions

We have analysed a simple extension of the SM in which light neutrino masses are linked

to the presence of n extra generations with both left- and right-handed neutrinos. The

Yukawa neutrino matrices are rank n, so if we do not impose lepton number conservation

and allow for right-handed neutrino Majorana masses, at tree level there are 2n massive

Majorana neutrinos and three massless ones. In order to obtain heavy neutrino masses

above the experimental limits from direct searches at LEP, the Dirac neutrino masses

should be at the electroweak scale, similar to those of their charged lepton partners, and

the right-handed neutrino Majorana masses can not be too high (of order 10 TeV at most).

The three remaining neutrinos get Majorana masses at two loops, therefore this framework

provides a natural explanation for the tiny masses of the known SM neutrinos. On the

other hand, it should be kept in mind that the two-loop contribution to the neutrino mass

matrix is always present in this type of SM extensions, therefore the experimental upper

– 21 –

JHEP07(2011)122

limit on the absolute light neutrino mass scale leads to a relevant constraint which has to

be taken into account.

We have shown that the minimal extension with a fourth generation can not fit si-

multaneously the ratio of the solar and atmospheric neutrino mass scales, ∆m2sol/∆m2

atm,

the lower bound on the heavy neutrino mass from LEP and the limits on the mixing be-

tween the fourth generation and the first three from low energy universality tests. Then,

there are two possibilities: either enlarge the Higgs sector [47] or consider a five generation

extension. In this work we have analyzed the second one, while the first will be studied

elsewhere [74]. Notice that five generations are still allowed by the combination of collider

searches for its direct production, indirect effects in Higgs boson production at Tevatron

and LHC, and precision electroweak observables [19], provided the Higgs mass is roughly

mH > 300 GeV. However they will be either discovered or fully excluded at LHC, making

our proposal falsifiable in the very near future.

Given the large number of free parameters in a five generation framework (10 neutrino

Yukawa couplings, 2 charged lepton masses and 2 right-handed neutrino Majorana masses),

we have considered a very simple working example assuming that i) the linear combinations

of left-handed neutrinos that get Dirac masses at tree level are orthogonal to each other,

ii) each extra generation left-handed neutrino couples only to one of the two right-handed

SM singlet states and iii) each extra generation charged lepton couples only to one linear

combination of the (tree-level) massless neutrinos. Then, we are left with 2 neutrino Yukawa

couplings yE, yF , 2 charged lepton masses mE, mF , 2 right-handed neutrino Majorana

masses maR, a = 4, 5, which characterize the amount of total lepton number breaking, and

two small parameters ǫ, ǫ′ which determine the mixing among the first three generations

and the new ones. Moreover, at leading order the two-loop neutrino masses m2, m3 depend

only on ǫ,m4R, yE ,mE and ǫ′,m5R, yF ,mF , respectively (see eq. (4.5)). Even in this over-

simplified case we are able to accommodate all current data, including the observed pattern

of neutrino masses and mixings both for normal and inverted hierarchy spectrum. A definite

prediction of the model (independent of the above simplifying assumptions) is that the three

light neutrinos can not be degenerate.

We have explored the parameter space regions able to generate the correct scale of

neutrino masses, ∼ 0.05 eV. We find that for typical values of mE,m4D (mF ,m5D) at the

electroweak scale, we need ǫ2m4R (ǫ′2m5R) . 1 keV to obtain the atmospheric mass scale

(see figure 4).

We have also studied the current bounds on the mixing parameters ǫ and ǫ′ from the

non-observation of LFV rare decays ℓα → ℓβγ, as well as from universality tests. All of them

are independent of the Majorana masses miR, since they conserve total lepton number.

Depending on the light neutrino mass spectrum (normal or inverted), the strongest bounds

come from µ → eγ and from universality tests in π decays. Combining the information from

both processes we can set independent limits on ǫ and ǫ′ which being quite conservative

are of the order of the few percent, ǫ . 0.03 and ǫ′ . 0.04.

Finally, we have analysed the phenomenological prospects of the model. With respect

to LFV signals, future MEG data will improve the limits on the ǫ’s by a factor of about

3 while, if expectations from µ–e conversion are attained the limits on the ǫ’s will be

– 22 –

JHEP07(2011)122

pushed to 10−3. This region of observable LFV effects corresponds to the pseudo-Dirac

limit, maR . 1 GeV, i.e., two pairs of strongly degenerate heavy neutrinos. In this regime,

they can only be discovered at LHC using pure Dirac neutrino signatures, which are more

difficult to disentangle from the background.

On the other hand, we find that in the complementary region of very small mixing

ǫ, ǫ′ ≪ 10−3, maR & 1 GeV, the lighter Majorana neutrinos ν4, ν4 will lead to observable

same-sign di-lepton signatures at LHC. A detailed study is missing, but previous results

seem to indicate that a lower bound on m4 of order 300 GeV could be set with 5 fb−1 of

LHC data at 10 TeV [70].

Acknowledgments

We thank M. Hirsch and E. Fernandez-Martınez for useful discussions. This work has

been partially supported by the Spanish MICINN under grants FPA-2007-60323, FPA-

2008-03373, Consolider-Ingenio PAU (CSD2007-00060) and CPAN (CSD2007-00042), by

Generalitat Valenciana grants PROMETEO/2009/116 and PROMETEO/2009/128 and by

the European Union within the Marie Curie Research & Training Networks, MRTN-CT-

2006-035482 (FLAVIAnet). A.A. and J.H.-G. are supported by the MICINN under the

FPU program.

A The neutrino mass two-loop integral

The relevant integral is

J ≡ J(m4,m4,mα,mβ,mW ) =

=

∫

pq

p · q(

(p+q)2−m24

)(

(p+q)2−m24

)

(p2−m2α)(q2−m2

β)(p2−m2W )(q2−m2

W ), (A.1)

where∫

pq=

∫∫

d4p

(2π)4d4q

(2π)4.

We combine propagators with the same momentum by using

1

(p2 − m2α)(p2 − m2

W )=

1

(m2α − m2

W )

∫ m2α

m2

W

dt1(p2 − t1)2

,

then

J =

∫

t

∫

pq

(pq)

(p2 − t1)2(q2 − t2)2((p + q)2 − t3)2,

where∫

t=

1

(m2α − m2

W )

1

(m2β − m2

W )

1

(m24 − m2

4)

∫ m2α

m2

W

dt1

∫ m2

β

m2

W

dt2

∫ m2

4

m24

dt3 .

Now we use the standard Feynman parametrization to combine the last two propagators

and perform the integral in q, which leads to

J = − i

(4π)2

∫

t

∫ 1

0

dx

1 − x

∫

p

p2

(p2 − t1)2(

p2 − t3/(1 − x) − t2/x)2 .

– 23 –

JHEP07(2011)122

The integral in p can be reduced by using an additional Feynman parameter and the final

result can be written as

J = − 2

(4π)4

∫ 1

0dx

∫ 1

0dy

∫

t

y(1 − y)x

t3xy + t2(1 − x)y + t1x(1 − x)(1 − y). (A.2)

The integrals in t1, t2, t3 can be done analytically and reduced to logarithms. The expres-

sions obtained are complicated but can be used to feed the final numerical integration in

x and y which converges smoothly for most of the parameters. The expression in (A.2) is

also very useful to obtain different approximations for small masses as compared with the

largest mass in the integral. For that purpose one can use

lima→0

limb→a

1

b − a

∫ b

adtf(t) = lim

a→0f(a) = f(0) .

Thus, for instance if m4,m4 ≫ mα,mβ,mW ∼ 0 we can take t1, t2 → 0 in the integrand

and perform trivially the remaining integrals,

J = − 1

(4π)41

m24− m2

4

lnm2

4

m24

, (A.3)

in agreement with the result in [48].

If mβ,m4,m4 ≫ mα,mW ∼ 0 the integral can also be computed (take t1 → 0 in the

integrand and perform the rest of the integrals). The result can be written in terms of the

dilogarithm function Li2(x) and it is rather compact,

J = − 1

(4π)4m2β

(

π2

6−

m24

m24− m2

4

(

Li2

(

1 −m2

β

m24

)

− m24

m24

Li2

(

1 −m2

β

m24

)))

.

When mβ → 0 it reduces, as it should, to (A.3).

We are especially interested in the case mα = mβ ≡ mE with mE > mW , but m4,m4

could be larger or smaller than mE (and even smaller than mW since we only know that

m4 ≥ m4 > 63 GeV). Some asymptotic expressions can be obtained when there are large

hierarchies in masses

J ≈ − 1

(4π)41

m2X

lnm2

X

m2Y

,

where mX is the heaviest of mE ,m4,m4,mW and mY the next to the heaviest of these

masses. This expression can be used to perform analytical estimates but, since in the

allowed range of masses the hierarchies cannot be huge, we do expect large corrections to

these estimates. Fortunately, as commented above, the exact value of J for all values of

the masses can be obtained numerically rather easily using (A.2). For fast estimates one

can use

J ≈ 1

(4π)41

m24− m2

4 − m2E

ln

(

m24 + m2

E

m24

)

, mE,m4,m4 ≫ mW ,

which interpolates smoothly the different asymptotic expressions and reproduces the com-

plete result with an error less than 50% in the worse case.

– 24 –

JHEP07(2011)122

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ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

On the annual modulation signal in

dark matter direct detection

Juan Herrero-Garcia,a,b Thomas Schwetza and Jure Zupanc,1

aMax-Planck-Institut fur Kernphysik,PO Box 103980, 69029 Heidelberg, Germany

bDepartamento de Fisica Teorica, and IFIC, Universidad de Valencia-CSIC,Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, Spain

cDepartment of Physics, University of Cincinnati,Cincinnati, Ohio 45221, U.S.A.

E-mail: [email protected], [email protected], [email protected]

Received December 15, 2011Revised February 8, 2012Accepted February 13, 2012Published March 5, 2012

Abstract. We derive constraints on the annual modulation signal in Dark Matter (DM)direct detection experiments in terms of the unmodulated event rate. A general boundindependent of the details of the DM distribution follows from the assumption that themotion of the earth around the sun is the only source of time variation. The bound is validfor a very general class of particle physics models and also holds in the presence of an unknownunmodulated background. More stringent bounds are obtained, if modest assumptions onsymmetry properties of the DM halo are adopted. We illustrate the bounds by applyingthem to the annual modulation signals reported by the DAMA and CoGeNT experiments inthe framework of spin-independent elastic scattering. While the DAMA signal satisfies ourbounds, severe restrictions on the DM mass can be set for CoGeNT.

Keywords: dark matter theory, dark matter experiments, dark matter detectors


1On leave of absence from University of Ljubljana, Depart. of Mathematics and Physics, Jadranska 19,1000 Ljubljana, Slovenia and Josef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.

c© 2012 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2012/03/005

JCAP03(2012)005

Contents

1 Introduction 1

2 Notation 2

3 General bound on the annual modulation amplitude 4

4 Bounds on the annual modulation for symmetric halos 6

5 Applying the bound to data 9

5.1 Single target detector 95.1.1 CoGeNT 11

5.2 Bounds for multi-target experiments 165.2.1 DAMA 17

6 Summary and discussion 18

A Maxwellian halo 21

B Translating bounds in vm space to observable quantities 22

C Alternative procedure for optimizing the bound in presence of unknown

background 25

1 Introduction

A smoking-gun signature of the dark matter (DM) signal in DM direct detection experimentsis the annual modulation of the event rate. This arises because the earth rotates around thesun, while at the same time the sun moves relative to the DM halo [1, 2]. Currently, twoexperiments report annual modulation of the signals, DAMA with 8.9σ significance [3], andCoGeNT with a significance of 2.8σ [4]. An important question is whether the observedmodulation rates are consistent with the interpretation that the signal is due to DM. Ideally,one would like to answer this question without requiring a detailed knowledge of the DMhalo. As we will show below, we come very close to this ideal.

The DM scattering rate in the detector is determined by the particle physics properties ofDM and by the astrophysical properties of the DM halo. The latter carry large uncertainties.When presenting results of DM direct detection experiments, it has become customary touse a “standard” Maxwellian DM velocity distribution in order to show the constraints onDM mass and scattering cross section. This is very likely an oversimplification, with N-bodysimulations indicating a more complicated structure of the DM halo, see e.g., [5]. It is wellknown that the annual modulation signal is sensitive to such deviations from the Maxwellianhalo [6–8]. DM streams are particularly relevant for the annual modulation signal and canlead to large effects. These can, for instance, be comparable in size to the modulation effectdue to the whole Maxwellian halo [9–12].

In the following we will derive constraints on the amplitude of the annual modulationusing measured unmodulated rates. The derivation of these bounds is made possible by

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JCAP03(2012)005

the fact that the velocity of the earth rotating around the sun, ve ∼ 30 km/s, is muchsmaller than the typical velocity of DM particles in the halo, 〈v〉 ∼ 200 km/s, and we canexpand the velocity integral to first order in ve. The derived constraints on the annualmodulation then serve as a consistency check for the hypothesis that the annual modulationis due to the DM signal. The check that we provide is (almost) independent of DM haloproperties. Previously halo independent methods to interpret DM direct detection data havebeen developed and applied in [13–18], with different goals than ours. In ref. [14] a method fordetermining the DMmass independently of the halo was developed, while refs. [13, 18] focusedon extracting halo properties from the data. Compatibility studies of different experimentswithout adopting assumptions on the halo have been performed in [15–17]. Our goal is toobtain a consistency check between the modulation amplitude and the unmodulated signalwithin a given experiment.

The structure of the paper is as follows: in section 2 we set up the notation, followedin section 3 by a derivation of the general bound on the modulation signal under the verymild assumption that the properties of the DM halo in our vicinity do not change on timescales of months. In section 4 we impose further symmetry requirements on the DM velocitydistribution and obtain successively more stringent bounds. In section 5 we then demonstratethe usefulness of our bounds by applying them to the modulation signals reported in CoGeNTand DAMA within elastic spin-independent scattering. We find that our method leads to non-trivial restrictions on DM interpretations of the modulation signal in CoGeNT whereas theDAMA signal satisfies our bounds for the relevant DM masses. We summarize in section 6.In appendix A we consider the Maxwellian DM halo and show that the expansion in ve israther accurate in this case. Furthermore, we illustrate the various bounds compared to themodulation signal expected for the Maxwellian halo. Appendix B provides technical detailsnecessary to translate the bounds derived for the halo velocity integral into observable eventrates. Supplementary material related to the treatment of an unknown background is givenin appendix C.

2 Notation

The differential rate in events/keV/kg/day for DM χ to scatter off a nucleus (A,Z) anddepositing the nuclear recoil energy Enr in the detector is

R(Enr, t) =ρχmχ

1

mA

∫

v>vm

d3vdσAdEnr

vfdet(v, t). (2.1)

Here ρχ ≃ 0.3GeV/cm3 is the local DM density, mA and mχ are the nucleus and DM masses,σA the DM-nucleus scattering cross section1 and v the 3-vector relative velocity between DMand the nucleus, while v ≡ |v|. For a DM particle to deposit recoil energy Enr in the detectora minimal velocity vm is required, restricting the integral over velocities in (2.1). For elasticscattering it is

vm =

√

mAEnr

2µ2χA

, (2.2)

where µχA is the reduced mass of the DM-nucleus system.

1Throughout this work we assume that DM is dominated by a single particle species. A generalization tomulti-component DM is straightforward and will be pursued in future work.

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JCAP03(2012)005

The function fdet(v, t) describes the distribution of DM particle velocities in the detectorrest frame with fdet(v, t) ≥ 0 and

∫

d3vfdet(v, t) = 1. It is related to the velocity distributionin the rest frame of the sun by

fdet(v, t) = fsun(v + ve(t)) , (2.3)

where ve(t) is the velocity vector of the earth, which we write as [9]

ve(t) = ve[e1 sinλ(t)− e2 cosλ(t)] (2.4)

with ve = 29.8 km/s, and λ(t) = 2π(t − 0.218) with t in units of 1 year and t = 0 atJanuary 1st, while e1 = (−0.0670, 0.4927,−0.8676) and e2 = (−0.9931,−0.1170, 0.01032)are orthogonal unit vectors spanning the plane of the earth’s orbit, assumed to be circular.Similarly, the DM velocity distribution in the galactic frame is connected to the one in therest frame of the sun by fsun(v) = fgal(v + vsun), with vsun ≈ (0, 220, 0) km/s + vpec andvpec ≈ (10, 13, 7) km/s the peculiar velocity of the sun. We are using galactic coordinateswhere x points towards the galactic center, y in the direction of the galactic rotation, and ztowards the galactic north, perpendicular to the disc. As shown in [7], eq. (2.4) provides anexcellent approximation to describe the annual modulation signal.

In the following we consider the typical situation, where the differential cross section isgiven by

dσAdEnr

=mA

2µ2χAv

2σ0AF

2(Enr) , (2.5)

where σ0A is the total DM-nucleus scattering cross section at zero momentum transfer, and

F (Enr) is a form factor. The event rate is then given by

R(Enr, t) = C F 2(Enr) η(vm, t) with C =ρχσ

0A

2mχµ2χA

(2.6)

and the halo integral

η(vm, t) ≡∫

v>vm

d3vfdet(v, t)

v. (2.7)

Here and in the following vm and Enr are related by eq. (2.2). This formalism covers a widerange of possible DM-nucleus interaction models, including the standard spin-independentand spin-dependent scattering. The results we derive below apply to all the cases wheredσA/dEnr ∝ 1/v2, in which case the halo integral (2.7) is obtained. The arguments caneasily be generalized to a non-trivial q2 dependence of the interaction, which would introducean additional Enr dependent function in (2.6) but would not change (2.7). Our results donot apply, however, to a non-standard velocity dependence of the cross section, which wouldmodify eq. (2.7).

The rate will have a time independent component and an annually modulated compo-nent which we define as

R(Enr, t) = R(Enr) + δR(Enr, t) = C F 2(Enr) [η(vm) + δη(vm, t)] . (2.8)

Below we will be specifically interested in purely sinusoidal time dependence with period ofone year, in which case we can write

δR(Enr, t) = AR(Enr) cos 2π[t− t0(Enr)] ,

δη(vm, t) = Aη(vm) cos 2π[t− t0(Enr)] .(2.9)

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The peak of the annual modulation occurs at t0, which in general depends on Enr (or equiv-alently on vm). The modulation amplitudes of the event rate, AR, and of the halo integral,Aη, are related through AR(Enr) = CF 2(Enr)Aη(vm) ≥ 0. We will first derive bounds on Aη

in terms of the time averaged value of the halo integral η. In section 5 we will then translatethe bounds into constraints involving the observable quantities AR and R.

3 General bound on the annual modulation amplitude

Assumption 1. We assume that the only time dependence comes from ve(t) and there isno explicit time dependence in fsun.

This assumption implies that the halo is spatially constant at the scale of the sun-earthdistance and also constant in time on scale of months. Under this assumption we can derivea general bound on the annual modulation by expanding eq. (2.7) in ve/vm ≪ 1.2 Using forshort f ≡ fsun, we have

η(vm, t) =

∫

|v−ve|>vm

d3vf(v)

|v − ve|(3.1)

=

∫

v>vm

d3vf(v)

v+

∫

d3vf(v)v · ve(t)

v3[Θ(v − vm)− δ(v − vm)vm] +O(v2e/v

2m) ,

(3.2)

where the first term in (3.2) gives the time independent part of the DM scattering signal. Inpolar coordinates the time independent halo integral is then given by

η(vm) =

∫

v>vm

d3vf(v)

v=

∫

vm

dv v

∫ 2π

0dϕ

∫ 1

−1d cosϑ f(v, ϑ, ϕ) . (3.3)

The second term in (3.2) corresponds to the time dependent part of the DM scattering signal,with ve(t) given in (2.4). Expanding to linear order in ve thus leads to an annual modulationsignal that has a sinusoidal shape. One can check experimentally for the convergence ofthe expansion by searching for higher order terms, ∝ v2e sin

2[2π(t − t0)], etc. Note that theexpansion is in ve/v, where v & vm, so that the accuracy is typically better than O(ve/vm).In appendix A we demonstrate explicitly that this expansion is rather accurate in the caseof a Maxwellian halo.

By expanding the velocity integral in the small quantity ve we assume that f(v) issmooth on scales . ve. Hence, our bounds do not apply in situations where f(v) has strongstructures at scales smaller than ve. An example would be a very cold stream with velocityvstream and a dispersion smaller than ve. In such a case the expansion will not be accuratefor vm in the range |vm − vstream| . ve sinα, though it may still work to good accuracy forvm outside this range.

The time dependent component in (3.2) has two contributions

δη(vm, t) =

∫

d3vf(v)v · ve(t)

v3[Θ(v − vm)− δ(v − vm)vm] . (3.4)

2Typically vm is sensitive to both the nucleus and the DM mass. For a 10GeV DM mass and a recoilenergy Enr of a few KeV, we get for the nuclei we have analysed (Ge, Na, I) that vm & 5ve km/s, so theexpansion is rather accurate (as we have checked explicitly for a Maxwellian halo in appendix A).

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The term with the Θ-function comes from expanding the denominator in eq. (3.1) and involvesa volume integral over the region v > vm. The term with the δ-function comes from takinginto account the effect of ve on the integration boundary, and the δ-function indicates thatthe argument has to be evaluated at the surface v = vm. Let us treat the two terms separatelyand define

vg(vm)g(vm) ≡∫

d3vf(v)v

v3δ(v − vm) , (3.5)

vG(vm)G(vm) ≡∫

d3vf(v)v

v3Θ(v − vm) . (3.6)

The unit vectors vg(vm) and vG(vm) give the corresponding weight averaged DM wind di-rections in the earth’s rest frame. In general they point in different directions and candepend on vm. For the Maxwellian halo they point in the same direction and are equal tovg = vG = −vsun. We will treat such special cases in the next section. The positive functionsg(vm) and G(vm) are given by

g(vm) =

∫ 2π

0dϕ

∫ 1

−1d cosϑf(vm, ϑ, ϕ) cosϑ , (3.7)

G(vm) =

∫

vm

dv

∫ 2π

0dϕ

∫ 1

−1d cosϑ′f(v, ϑ′, ϕ) cosϑ′ , (3.8)

where ϑ(ϑ′) is the angle between v and vg(vG). The time dependent halo integral is thusgiven by

δη(vm, t) = ve(t) · [vG(vm)G(vm)− vg(vm)vmg(vm)] . (3.9)

As already mentioned, the form of ve(t) from eq. (2.4) implies that our approximationslead to strictly sinusoidal modulations, justifying the ansatz in eq. (2.9), δη(vm, t) =Aη(vm) cos 2π(t − t0). Using eq. (3.9), the modulation amplitude can be constrained inthe following way:

Aη(vm) ≤ ve[

vmg(vm) +G(vm)]

. (3.10)

Note that the two terms in eq. (3.9) will contribute to the modulation amplitude proportionalto sinαg and sinαG, respectively, where αg,G is the angle between vg,G and the directionorthogonal to the plane of the earth orbit, i.e., cosαg,G = vg,G · e3, with e3 = e1 × e2, ande1,2 given below eq. (2.4). In general αg,G will depend on vm. To derive the bound we haveassumed the maximal possible effect, corresponding to sinαg = sinαG = 1. Since we do notknow the relative sign of the contributions from g(vm) and G(vm) we have to take the sumof the moduli.

The function g(vm) is bounded from above by

g(vm) ≤∫

dϕ dcosϑf(vm, ϕ, ϑ)| cosϑ| ≤∫

dϕ dcosϑf(vm, ϕ, ϑ) = − 1

vm

dη

dvm, (3.11)

where the last equality follows from eq. (3.3). Note that η(vm) is a monotonously decreasingfunction and therefore, dη(vm)/dvm ≤ 0. Using again the last equality above, also a boundfor G(vm) can be derived:

G(vm) ≤ −∫

vm

dv1

v

dη

dv=

η(vm)

vm−∫

vm

dvη(v)

v2. (3.12)

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The equality in (3.12) follows from integration by parts. The inequalities (3.11) and (3.12) aresaturated if f(v) ∝ δ(ϑ), i.e., the hypothetical situation that all DM particles have the samedirection and their velocities have no transversal component. Using eqs. (3.11) and (3.12)the bound on the modulation amplitude (3.10) becomes

Aη(vm) ≤ ve

[

− dη

dvm+

η(vm)

vm−∫

vm

dvη(v)

v2

]

(Assumption 1) , (3.13)

where the first two terms on the r.h.s. are positive and the third is negative. Ifthe DM scattering rate R is measured, the r.h.s. is fully determined experimentally byη(vm) = R(Enr)/CF 2(Enr), and can be compared to the observed modulation throughAη = AR/CF 2(Enr), see section 5 for details. Note that the phase of the modulation (whichmay vary with vm) does not appear in the bound (3.13). The bound applies on the modula-tion amplitude, irrespective of the phase. Eq. (3.13) is one of the main results of this paper.The bound is rather general and holds under the very mild Assumption 1 specified above.For it to be saturated the DM velocity distribution at the position of the solar system wouldneed to be highly nontrivial. For instance, the bound in (3.10) can be saturated if there is aDM stream aligned with the ecliptic and it is strong enough so that it dominates the velocitydistribution at v = vm. Note that even in this case, for different vm the bound will not besaturated. To saturate in addition eq. (3.11) or (3.12), the stream should have no transversalvelocity dispersion. Any modulation signal which violates, or even just saturates, this boundis very unlikely to have a DM origin.

4 Bounds on the annual modulation for symmetric halos

Assumption 2. In addition to Assumption 1 we now assume that vg defined in eq. (3.5)is independent of vm, i.e., we assume that there is a constant direction vhalo ≡ vg governingthe shape of the DM velocity distribution in the sun’s rest frame.

From this assumption it follows immediately that vG is also constant and equal to vhalo,so that

g(vm) = − dG

dvm, (4.1)

and eq. (3.9) becomes

δη(vm, t) = −ve(t) · vhalo [vmg(vm)−G(vm)] . (4.2)

The crucial difference to the general case (3.9) is that we were able to pull the velocity vectorin front of the bracket. The functions vmg(vm) and G(vm) are both positive. Their relativesizes determine whether the bracket is positive or negative. For small vm the function G(vm)typically dominates and we get an extra half a year shift in the peak of the modulation (forthe Maxwellian halo the peak would then be in December). For larger vm the boundary termvmg(vm) dominates and the whole bracket is positive (and thus for the Maxwellian halo thepeak in this case is in June, see appendix A).

Assumption 2 is fulfilled if f(v) obeys certain symmetry requirements that we candeduce from eq. (3.5). For a given vm we chose a coordinate system such that vg = (1, 0, 0),and

∫

d3vf(v)vyδ(v − vm) = 0 ,

∫

d3vf(v)vzδ(v − vm) = 0 . (4.3)

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Assumption 2 implies that these relations hold for any vm in the same coordinate system.The distribution in vx (i.e., in the direction of vhalo) can be arbitrary, and can for instanceeven include several peaks, as long as vg does not flip sign. Therefore, we have to requirethat the integral over the half-sphere with vx > 0 is larger than the one with vx < 0 forall vm:

∫

vx<0d3vf(v)vxδ(v − vm) <

∫

vx>0d3vf(v)|vx|δ(v − vm) . (4.4)

One possibility to satisfy the condition (4.3) is a symmetric velocity distribution, withf(vx, vy, vz) = f(vx,−vy, vz) and f(vx, vy, vz) = f(vx, vy,−vz) for all vx. Eq. (4.3) can alsobe satisfied for distributions asymmetric in vy and/or vz, however, in this case the asymmetryhas to be such that the cancellation between vy,z > 0 and < 0 happens for all radii vm.

Assumption 2 is fulfilled for the standard Maxwellian halo, as well as for any otherisotropic velocity distribution. Up to small corrections due to the peculiar velocity of the sunit holds also for tri-axial halos, and covers also streams parallel to the motion of the sun, suchas a dark disc co-rotating with the galactic stellar disc [19]. Note that for all those examplesthe DM direction vhalo is aligned with the motion of the sun (up to the peculiar velocity thatleads to sub-leading corrections). Let us introduce this as an additional assumption:

Assumption 2a. In addition to Assumption 2 we require that the preferred direction vhalo

is aligned with the motion of the sun relative to the galaxy.

As mentioned above, for many realistic cases fulfilling Assumption 2 also this additionalrequirement is fulfilled. An exception would be the situation when the DM density at the sun’slocation is dominated by a single stream from an arbitrary direction and the contribution ofthe static halo is negligible.

Let us now use eq. (4.2) to derive a bound on the modulation. We have ve(t) · vhalo =−ve sinαhalo cos(t− t0), where t0 is now independent of vm. Here αhalo is the angle betweenvhalo and e1 × e2, i.e., the projection of vhalo on the ecliptic plane, with sinαhalo ≥ 0.Assumptions 2 and 2a thus imply that the phase of the modulation is independent of vm(and therefore independent of Enr). As discussed above this is up to a sign flip of the squarebracket in eq. (4.2) that can happen due to the two competing terms. To take this intoaccount we now define

δη(vm, t) = A′η(vm) cos(t− t0) , (4.5)

where t0 is constant and we allow a sign change for A′η(vm). This is different from Aη,

which has been defined to be positive in eq. (2.9). While for Assumption 2 the phase isarbitrary but constant, Assumption 2a also gives a prediction for the phase — that themaximum (or minimum) of the event rate is around June 2nd. This can be checked in theexperiment by looking at the annual modulation phase at different energy bins. Hence, fromthe experimental information on the phase we can conclude whether Assumptions 2 or 2amay be justified and whether it makes sense to apply the corresponding test.

A useful bound on the modulation can be obtained by first integrating eq. (4.2) over vm,

∫ vm2

vm1

dvmA′η(vm) = ve sinαhalo

∫ vm2

vm1

dvm[vmg(vm)−G(vm)]

= ve sinαhalo [vm1G(vm1)− vm2G(vm2)] ,

(4.6)

where the second equality is obtained by integrating vmg(vm) by parts using g(vm) =−dG/dvm, eq. (4.1). Note that the A′

η and αhalo are defined in such a way that A′η is

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JCAP03(2012)005

positive for large vm, above the last sign flip. Experimentally, this is at present the mostinteresting region of the parameter space. Both putative modulation signals, at CoGeNTand DAMA, have a peak close to June, and no half year phase change is seen within theobserved energy range.3

In general we do not know which of the two vmG(vm) terms in (4.6) dominates. Droppingthe smaller of the two and using the bound (3.12) we arrive at

∣

∣

∣

∣

∫ vm2

vm1

dvmA′η(vm)

∣

∣

∣

∣

≤ ve∣

∣sinαhalo

∣

∣max

[

η(vm1)− vm1

∫

vm1

dvη(v)

v2, vm1 → vm2

]

. (4.7)

Assumption 2 allows for arbitrary directions of the DM wind, therefore we need to use| sinαhalo| → 1 above, while for Assumption 2a one has sinαhalo ≃ 0.5. Note that a signflip of A′

η will lead to cancellations in the integral on the l.h.s. of eq. (4.7) and make thebounds weaker. An observation of such a sign flip in the modulation amplitude would be astrong experimental evidence that the modulation is due to a DM signal. If such a sign flipis observed, one might be able to obtain stronger constraints by applying the bound for theregion below and above the sign flip separately, to avoid the cancellations.

At present there is no indication of such a sign flip in which case the bound (4.7)simplifies to

∫ vm2

vm1

dvmA′η(vm) ≤ ve

[

η(vm1)− vm1

∫

vm1

dvη(v)

v2

]


and∫ vm2

vm1

dvmA′η(vm) ≤ 0.5 ve

[

η(vm1)− vm1

∫

vm1

dvη(v)

v2

]

(Assumption 2a) . (4.9)

The bounds (3.13), (4.8), (4.9) are the central results of this paper. In the following wewill refer to them as

Eq. (3.13) “general bound” (Assumption 1),Eq. (4.8) “symmetric halo” (Assumption 2),Eq. (4.9) “symmetric halo, sinα = 0.5” (Assumption 2a).

The term “symmetric halo” should be understood in the sense of eqs. (4.3) and (4.4).For completeness let us also mention an unintegrated bound, even though we will not

use it for the numerical analysis. Using eq. (4.2) we have for the modulation amplitude

Aη(vm) = ve∣

∣ sinαhalo

∣

∣

∣

∣vmg(vm)−G(vm)∣

∣ . (4.10)

Note that the minus sign between the two terms is conserved, while in the general case,eq. (3.10), one is forced to sum the two terms in the bound. From eqs. (3.11) and (3.12) onethen obtains the following bound on the modulation

Aη(vm) ≤ ve| sinαhalo|max

[ ∣

∣

∣

∣

dη

dvm

∣

∣

∣

∣

,η(vm)

vm−

∫

vm

dvη(v)

v2

]

. (4.11)

A similar bound has been obtained recently in [18]. If eq. (B4) of [18] is expanded in u ourbound (4.11) is obtained, if the derivative term dominates. However, eq. (B4) of [18] seems

3Note that the lowest energy bin in DAMA shows a somewhat smaller modulation amplitude, which mightbe an indication of a phase shift below the threshold.

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JCAP03(2012)005

to neglect the possibility that the second term in eq. (4.11) dominates. This assumption isjustified if data show no phase flip in the modulation and correspond to vm above the lastphase flip (A′

η ≥ 0). In this case eq. (4.11) becomes

A′η(vm) ≤ −ve sinαhalo

dη

dvm. (4.12)

Note that this bound is complementary to the ones from eqs. (4.8), (4.9). By taking theintegral over the modulation amplitude those bounds probe global properties of the amplitudeover the considered energy range, whereas eq. (4.12) bounds the local size of the modulationamplitude at a given value of vm. Both bounds are necessary conditions which a modulationsignal with a DM origin has to fulfill.

5 Applying the bound to data

In this section we show how the bounds (3.13), (4.8), (4.9) obtained in vm space for halointegrals can be applied to observable quantities, the unmodulated rate R and the amplitudeof the modulation of the rate, AR. The halo integral and the DM scattering rate are propor-tional to each other, see eq. (2.6). The relation is complicated by the fact that experimentstypically do not observe the recoil energy Enr directly. The nuclear recoil energy is relatedto the observed energy through a quenching factor Q. In the case of CoGeNT and DAMAthe observed energy Eee is measured in electron equivalent and is related to the recoil energyby Q = Eee/Enr. In general this is a nonlinear equation, as Q also depends on the recoilenergy. Finally, in an experiment data is reported in bins, and the continuous bounds derivedabove have to be integrated over the bin sizes. We relegate the details of the derivation toappendix B, and quote below only the final results.

5.1 Single target detector

Let us first assume that the target consists of a single material, as is for instance the casefor CoGeNT. We denote the modulation amplitude and the unmodulated rate in bin i as Ai

and Ri, respectively, both in units of counts/day/kg/keVee. The general bound (3.13) thenbecomes

Ai ≤ ve

Ri(αi + βi)−Ri+1α′i+1 − 〈κ〉i

N∑

j=i

Rjγj


and the bounds for the symmetric halo are

N∑

j=i

Ajxj ≤ ve sinα

Riyi − 〈vm〉iN∑

j=i+1

Rjγj

(Assumptions 2, 2a) , (5.2)

where the bounds (4.8) (Assumption 2) and (4.9) (Assumption 2a) are obtained for sinα = 1and 0.5, respectively. The bin index i runs from 1 to N , while the rates Ri are zero for i > N .The coefficients αi, α

′i, βi, 〈κ〉i , γi, xi, yi, 〈vm〉i are given in eqs. (B.9) and (B.12). They are

known quantities calculable in terms of the form factor F (Enr), quenching factor Q(Enr),and Jacobians needed for changing the variable from vm to Eee. They depend on the DMmass mχ via the reduced mass µχA, eq. (2.2), needed to convert vm into Enr. The dependence

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JCAP03(2012)005

on the scattering cross section and the local DM density is encoded in the constant factorC, eq. (2.6). This factor cancels completely, as expected, since it is a common factor for themodulation as well as for the rate.

The rates Ri in the bounds (5.1) and (5.2) are the unmodulated scattering rate inducedby DM without including any backgrounds. In a “background free” experiment, where thefull observed event rate is due to DM, the bounds can be applied as they are. Here wewant to be more conservative, and consider also the situation where an unknown backgroundmay contribute to the unmodulated rate. The remaining assumption is then just that thebackground itself is not modulated, only the DM signal. In each bin a fraction ωi of theobserved count rate Ri is due to DM, i.e.,

Ri = Riωi , 0 ≤ ωi ≤ 1 , (5.3)

so that we can replace Ri → Riωi in eqs. (5.1) and (5.2) to obtain,

Ai ≤ ve

Riωi(αi + βi)−Ri+1ωi+1α′i+1 − 〈κ〉i

N∑

j=i

Rjωjγj


N∑

j=i

Ajxj ≤ ve sinα

Riωiyi − 〈vm〉iN∑

j=i+1

Rjωjγj

(Assumptions 2, 2a) . (5.5)

Now we have to find a set of ωi (i = 1, . . ., N), such that the bound becomes the “weakest”.In the following we describe two different procedures for this task. Procedure 1 is easy toimplement but gives slightly weaker bounds, whereas procedure 2 involves an optimizationalgorithm but gives somewhat more stringent bounds.

Procedure 1. We can sum eq. (5.4) from bin i to N and drop the last term in the squarebracket, since 〈κ〉i and γi are positive:

N∑

j=i

Aj ≤ ve

Riωiαi +N∑

j=i

Rjωjβj

≤ ve

Riαi +N∑

j=i

Rjβj


where the last inequality was obtained by setting ωi = 1. Similarly we can drop the secondterm from eq. (5.5) and obtain

N∑

j=i

Ajxj ≤ ve sinαRiωiyi ≤ ve sinαRiyi (Assumptions 2, 2a) , (5.7)

where the last inequality holds for yi > 0.4 These bounds have to be satisfied for all bins i.

Procedure 2. A somewhat stronger bound can be obtained by searching for the optimalchoice for the ωi, without dropping the last terms in eqs. (5.4) and (5.5). We present herea method based on a least-square minimization (an alternative procedure is outlined in ap-pendix C). Let us denote the r.h.s. of eqs. (5.4) and (5.5) as Bi which are functions of ωj .Then we can construct from eq. (5.4) the following least-square function

X2 =N∑

i=1

(

Ai −Bi

σAi

)2

Θ(Ai −Bi) (Assumption 1) , (5.8)

4According to eq. (B.12) yi can also become negative. In that case the bound (5.7) is always violated forany ωi ≥ 0.

– 10 –

JCAP03(2012)005

where σAi is the 1σ error on Ai (errors on Bi are typically much smaller and we neglect them

here). This X2 can now be minimized with respect to ωj under the condition 0 ≤ ωj ≤ 1.The Θ function takes into account that there is only a contribution to X2 if the bound isviolated. Hence, X2 will be zero if the bound is satisfied for all bins. A non-zero value ofX2 indicates that the bound is violated for some bin(s), weighted by the corresponding errorin the usual way. In the case of eq. (5.5) one has to take into account that the modulationamplitude in each bin is used several times, leading to correlated errors for the l.h.s.:

X2 =N∑

i,j=1

(Ai −Bi)S−1ij (Aj −Bj)Θ(Ai −Bi)Θ(Aj −Bj) (Assumption 2,2a) , (5.9)

where

Ai ≡N∑

k=i

Akxk and Sij =N∑

k=1

dAi

dAk

dAj

dAk

(

σAk

)2=

N∑

k=max(i,j)

(

xkσAk

)2. (5.10)

While non-zero values of the X2 functions (5.8) and (5.9) can be considered as a qual-itative measure for the violation of the bound, the precise distribution of them should bedetermined by Monte Carlo studies. From the definition one can expect, however, that theywill be approximately χ2 distributed if the bound is violated.

5.1.1 CoGeNT

Let us consider now the modulation signal reported by the CoGeNT experiment at 2.8σ [4].It has been pointed out that for specific assumptions on the halo (standard Maxwellian) thereis a tension between the modulated and unmodulated rate in CoGeNT [20, 22, 23]. Recentanalyses on the CoGeNT modulation signal can be found in refs. [24, 25], see also [18].Here we use the CoGeNT data for a case study and apply the above bounds assumingspin-independent elastic scattering. For the germanium quenching factor we use Eee[keV] =0.199(Enr[keV])1.12 [26]. We adopt the Helm parameterization of the spin-independent formfactor, F (Enr) = 3e−q2s2/2[sin(qr) − qr cos(qr)]/(qr)3, with q2 = 2mAEnr and s = 1 fm,r =

√R2 − 5s2, R = 1.2A1/3 fm. We use the data from figure 6 of ref. [20] where the total

rate R and the modulation amplitude AR are given in four bins of Eee between 0.5 and3.1 keV. We reproduce the data in table 1.5

The modulation amplitude has been extracted in three different ways in [20]. First,by fitting independently the modulation phase for each energy bin, second, by assuming aconstant phase for all bins, and third, by fixing the phase such that the modulation maximumis on June 2nd. These are precisely the requirements corresponding to our Assumptions 1,2, 2a, respectively. We can thus use the appropriate data on the modulation for each ofthe three assumptions. The modulation amplitudes in the first and the second case are verysimilar, while in the third case (forcing the phase to equal June 2nd) the amplitudes arelower and error bars are larger.

Choosing three values of the DM mass as examples, we show in figure 1 the boundon the integrated modulation amplitude for Assumption 1, and using procedure 1. The binlabels on the horizontal axis give the ith energy bin, which is the lower limit of summationin eq. (5.6). The data on amplitudes are shown as red dots, while the bounds are shown in

5We thank Mariangela Lisanti for providing us the data from figure 6 of ref. [20].

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JCAP03(2012)005

Eee bins [keV] Mod. (Ass. 1, 2) Mod. (Ass. 2a) Unmod. rate Corrected rate

0.5–0.9 1.41± 0.79 0.90± 0.72 12.33± 0.52 5.29± 0.52

0.9–1.5 0.84± 0.59 0.37± 0.55 4.33± 0.39 3.36± 0.39

1.5–2.3 0.46± 0.24 0.48± 0.22 2.76± 0.16 2.76± 0.16

2.3–3.1 0.66± 0.24 0.27± 0.23 2.83± 0.17 2.83± 0.17

Table 1. CoGeNT data on modulation amplitude and unmodulated rate [4] in cnts/day/kg/keV, asreported in . 6 of ref. [20]. The modulation has been extracted allowing for indpendent phases in eachbin (Ass. 1) and for a constant but arbitrary phase (Ass. 2) (which lead to very similar amplitudes),and by requiring the maximum at June 2nd (Ass. 2a). In the last column we show the preliminarysurface events corrected unmodulated rate [21].

æ

æ

ææ

ì

ìì ì

ì

ì

ì ì

ì

ì

ì ì

Red dot æ : integrated modulation

Black diamond ì : general bound

(from bottom mΧ= 7, 10, 20 GeV)

0.5-0.9 0.9-1.5 1.5-2.3 2.3-3.10

2

4

6

8

10

12

Eee HKeVL

Cou

nts�Hk

eVkg

dayL

Figure 1. Procedure 1 upper bound compared to the integrated modulation amplitude from theCoGeNT data, from figure 6 of [20]. The red dots correspond to the l.h.s. of eq. (5.6). The blackdiamonds correspond to the upper bound obtained under Assumption 1, r.h.s. of eq. (5.6), for DMmasses of mχ = 7, 10, 20GeV (from bottom to top) . The bins on the horizontal axis indicate the bini from which we start to sum the data. Error bars correspond to 1σ.

black. Whenever a red dot lies above one of the bounds (within errors), the DM hypothesisis disfavoured. In figure 1 this happens for the case of DM mass mχ = 7GeV. Such lightDM thus cannot be the source of the modulation, even under the very general assumptionon the DM halo adopted here.

We see from figure 1 that the strongest restriction comes from the bins i = 2 or 3. Infigure 2 we show the constraints on DM mass that follow from the bounds on the modulationamplitudes in these two bins. The bounds are obtained from procedure 1 for the threeAssumptions 1, 2, 2a. We observe that the bound becomes stronger for smaller DM masses.This behaviour can be understood from how the coefficients defined in appendix B depend onthe reduced mass µχA. The bounds on amplitudes are violated for DM masses below 10, 27,33GeV, respectively, for the three assumptions. The lower bounds on mχ are summarized intable 2 (left part), where we also give the bounds at 1σ and 2σ. Notice that due to the smallermodulation amplitudes and larger errors when extracted from the data under Assumption 2a,at 95% CL the corresponding bound becomes weaker than the one from Assumption 2 andequal to the one from Assumption 1, although naively one would expect the opposite. Thiscan be traced back to the fact that, under Assumption 2a, the modulation phase is forced

– 12 –

JCAP03(2012)005

Figure 2. Bounds on the CoGeNT modulation amplitude for the three assumptions about the DMhalo (1, 2, 2a), and using eqs. (5.6) and (5.7) (procedure 1). We show the l.h.s. (integrated modulationamplitude) minus the r.h.s. (upper bound) divided by the error on the l.h.s., as a function of DMmass. We sum the data starting from bin i = 3, 2 and 3 for Assumptions 1, 2 and 2a, respectively.The bounds are violated in the regions where the curves are above zero, which are shaded in the plot.

unmodulated rate from [4] corrected unmod. rate [21]

Mean 68% 95% X2 ≤ 1 Mean 68% 95% X2 ≤ 1

Ass. 1: general bound 8.5 6 3 7.3 10 6.5 3 10

Ass. 2: symmetric halo 24 14 6 18 43 25 12.5 37

Ass. 2a: sym. halo, sinα = 0.5 27.5 13.5 3.5 16 59.5 23 3 35

Table 2. Lower bounds on the DM mass in GeV, from the requirement that the modulationamplitude in CoGeNT is consistent with the upper bound from the unmodulated rate, accordingto the Assumptions 1, 2, 2a on the DM distribution. The bounds are obtained from procedure 1,requiring that eqs. (5.6) or (5.7) are satisfied for the mean value, or the 68% and 95% CL limits. Thebounds labeled “X2 ≤ 1” are obtained from procedure 2 by requiring that X2 defined in eqs. (5.8)or (5.9) is less than 1. In the left part of the table we use the published unmodulated event ratesfrom [4], whereas for the right part of the table we adopt the preliminary results on surface eventscontamination at low energies from [21].

to take the value of June 2nd which is not the one preferred by the data. The extractedmodulation signal then gets weaker and consequently the bounds are more easily satisfied.We have also checked that bounds for Assumptions 2, 2a derived from eq. (4.12) give alwaysweaker limits than the ones discussed here, which are based on eq. (4.8) and eq. (4.9).

In figure 3 we show the X2 functions according to procedure 2 as a function of theDM mass. For a given value of mχ we minimize X2 numerically with respect to the ωj .The conclusion is similar to procedure 1, leading to similar lower bounds on the DM mass.Requiring that X2 ≤ 1 one finds mχ ≥ 7.3, 18, 16GeV for Assumptions 1, 2, 2a, respectively.We observe again the unusual situation that Assumption 2a leads to a weaker constraint, due

– 13 –

JCAP03(2012)005

0 10 20 30 40 50 60 70 80Dark matter mass m

χ [GeV]

0

5

10

15

X2

Assumption 1 (general)

Assumption 2 (symmetric)

Assumption 2a (sym. sinα = 0.5)

Figure 3. Bounds on the CoGeNT modulation amplitude for the three assumptions about the DMhalo (1, 2, 2a), always using procedure 2. We show X2 defined in eqs. (5.8) and (5.9) as a functionof DM mass. The thin dashed curve is for illustrative purpose only; it corresponds to the data anderrors on the modulation amplitude extracted with a free phase (Assumption 2) but using the boundfor Assumption 2a, see text for details. If we assume that X2 is distributed as a χ2 with 1 d.o.f. (seebelow) X2 = 1 (4) corresponds to 68% (95%) CL.

to the less significant signal for the modulation. To illustrate this effect we show in figure 3with a thin-dashed curve also X2 that would follow from the hypothetical situation wherethe modulation amplitude would be as strong as in the case of Assumption 2. That is, weallow for an arbitrary energy independent phase, but assume that this phase turned out tobe June 2nd (in contrast to the real CoGeNT data), so that we could apply the bound fromAssumption 2a to these data. We see that in this case one would obtain a much strongerbound, disfavoring DM masses up to 60GeV. This example shows the potential of ourmethod in cases of a strong modulation signal in the data. If the signal for the modulationitself is weak, the bounds we derived will also give only weak constraints.

A recent re-analysis of CoGeNT data indicates that a significant fraction of the eventexcess at low energies could be due to surface events [21]. Assuming that surface events arenot modulated, the unmodulated rate will be reduced after subtructing the surface events,while the modulation signal will remain, leading to a strengthening of our bounds. Here weestimate this effect by using the preliminary result for a surface events rejection efficiencyshown on slide 19 of the presentation in [21] (red curve). Averaging this curve for the binsused in our analysis we find that the unmodulated rate in the first and second bins arereduced by a factor 0.43 and 0.78, respectively, while the other two bins are not effected.This leads to the reduced event rates shown in the last column of table 1. Figure 4 showsthe plots equivalent to figures 2 and 3 but using the surface events corrected unmodulatedrate. The general bound remains essentially the same. In this case the strongest limit (fromprocedure 1) for the uncorrected rate comes from summing data starting from bin 3, whereasin the surface events corrected case it comes from bin 2, with only a minor reduction (79%)

– 14 –

JCAP03(2012)005

0 10 20 30 40 50 60 70 80Dark matter mass m

χ [GeV]

0

5

10

15

X2

Assumption 1 (general)

Assumption 2 (symmetric)

Assumption 2a (sym. sinα = 0.5)

Figure 4. Bounds on the CoGeNT modulation amplitude for the preliminary surface events correctedunmodulated rate [21] for the three assumptions about the DM halo (1, 2, 2a). The left plot showsthe bounds according to procedure 1, whereas the right plot shows X2 defined in eqs. (5.8) and (5.9)(procedure 2). The thin curves in the right panel are the bounds without surface event subtructionreproduced from figure 3 for the purpose of comparison with the thick curves obtain with the surfaceevent corrected rates.

of the rate, leading to a very similar limit. In contrast, for Assumptions 2 and 2a significantlystronger bounds are obtained, with the limit coming now from summing from bin 1, witha 43% reduced rate. The surface events corrected bounds are summarized in the right partof table. 2.

Due to the non-standard definition of the X2 functions (5.8) and (5.9), involving the Θ-function, the actual distribution of them is not clear a priori. Therefore we have performeda Monte Carlo study in order to determine the distribution. For a given DM mass wefirst determine the optimal set of ωi by minimizing the X2. For those ωi we assume thatthe bound is saturated and we simulate a large number of pseudo-data taking the r.h.s. ofeqs. (5.4), (5.5) as mean value for a Gaussian with standard deviation given by the actualerror on the modulation amplitude.6 For each random data set we calculate the X2 valueand obtain therefore the distribution of X2 assuming that the bound is saturated. Then wecan compare the X2 of the actual data and calculate the probability of obtaining a X2 largerthan the observed one. Note that this procedure is conservative, since we assume that themean value of the random data is the bound itself. If the “true” mean values are smallerthan the bound, the distribution of X2 would be shifted towards smaller values, leading tomore stringent bounds.

We show the results of this calculation in figure 5. The probabilities obtained in thisway are in qualitative agreement with the numbers reported in table 2. Under assumptions2 and 2a, surface event corrected data is inconsistent with the DM hypothesis for any DMmass at the 97% and 90% CL, respectively. We find that if the bound is violated (i.e.,for small DM masses) the X2 distribution is close to a χ2-distribution with 1 d.o.f.. Largedeviations from a χ2-distribution occur if X2 is close to zero (i.e., for larger DM masses). In

6In the case of Assumptions 2 and 2a we proceed iteratively. Starting from eq. (5.5) for i = N we generateAN , and then simulate successively Ai−1. This is necessary to obtain the correct random properties of thel.h.s. of eq. (5.5).

– 15 –

JCAP03(2012)005

0 10 20 30 40 50 60DM mass [GeV]

0.001

0.01

0.1

1

1 -

CL

0 10 20 30 40 50 60DM mass [GeV]

0.001

0.01

0.1

1

1 -

CL

dashed: surface events corrected

Consistency of CoGeNT modulation with unmod. rate

assu

mpti

on 1

assumption 2a

assumption 2

Figure 5. The probability to obtain a X2 larger than the one obtained from CoGeNT data, asdetermined by Monte Carlo simulation. Dashed curves correspond to surface event corrected data.

this case the X2 distribution is strongly peaked at small values close to zero. This indicatesthat in such situations it is in most cases possible to find a set of ωi such that X2 becomeszero. This is the reason why non-trivial constraints are obtained under assumptions 2, 2a forsurface events corrected data at large masses, although the values of X2 are relatively small,compare figure 4.

5.2 Bounds for multi-target experiments

So far we have restricted the discussion to the situation where only one target nucleus ispresent, as for example Ge in CoGeNT. Let us now generalize our bounds to experimentswhere several elements are used as DM target. In this case both the modulation amplitudeas well as the unmodulated rate will receive contributions from each element: Ai =

∑

nAni

and Ri =∑

nRni , where n labels the different target elements and i energy bins. The

bounds from procedure 1 hold for each of the elements separately, where in general all thecoefficients appearing in the equations will depend on the nucleus type. Summing eqs. (5.6)and (5.7) over n and using the fact that for positive ai and bi, the following inequality holds,∑

i aibi ≤ (∑

i ai)(∑

j bj), we derive:

N∑

j=i

Aj ≤ ve

Ri

∑

n

αni +

N∑

j=i

Rj

∑

n

|βnj |


N∑

j=i

Aj minn(

xnj)

≤ ve sinαRi

∑

n

|yni | (Assumptions 2, 2a) . (5.12)

On the l.h.s. of the last inequality it is necessary to take the minimum of xnj between all thenuclei present in the target, for each bin. We used that αi and xj are positive, see appendix B.

It is possible to generalize also the X2 method to the case of multiple targets. For eachnucleus the bounds (5.4) and (5.5) apply separately. We define the amplitude from element n

– 16 –

JCAP03(2012)005

æ æ æ æ æ æ æ æ æ æ

ô ô ô ô ô ô ô ô ô ô

ôô ô

ô

ôô ô

ôô

ô

ôô ô

ô

ô

ôô

ôô

ô

DAMA: Na (Q = 0.3) + I (Q = 0.09)


Purple triangle ô : general bound

(from bottom mΧ = 3, 20, 50 GeV)

3 4 5 6

Eee HKeVL

Arb

itra

ryu

nit

s

æ æ æ æ æ æ æ æ æ æô

ôô ô ô

ô

ô

ô

ô

ô

ô ô ô ô ô ôô

ô

ô

ô

DAMA: Na (Q = 0.3) + I (Q = 0.09)


Purple triangle ô : Assump. 2 and 2a (mΧ = 3 GeV)

3 4 5 6

Eee HKeVL

Arb

itra

ryu

nit

s

Figure 6. Upper bound compared to the integrated modulation amplitude for DAMA data forAssumption 1 (left) and Assumptions 2 and 2a (right). We assume quenching factors q = 0.3 andq = 0.09 for Na and I, respectively. The red dots correspond to the l.h.s. of eqs. (5.11) and (5.12),and the purple triangles to the r.h.s.. In the left panel we show the bounds for DM masses ofmχ = 3, 20, 50GeV (from bottom), in the right panel for mχ = 3GeV. Bins are 0.5 keV wide and wesum all bins starting from bin i shown on the horizontal axis up to 7 keVee. Error bars are negligibleand are not shown for clarity. All dark matter masses are compatible with the modulation.

in the bin i as Ani = ǫni Ai, and similarly for the rate Rn

i = ωni Ri. Then one can construct aX2

similar to the one in the previous section, from l.h.s. minus r.h.s. of the following inequality:

ǫni Ai ≤ ve

Riωni (α

ni + βn

i )−Ri+1ωni+1α

ni+1 − 〈κn〉i

N∑

j=i

Rjωnj γ

nj

(Assumption 1) .

(5.13)

This X2 is minimized over ωni ≥ 0, ǫni ≥ 0 given the constraints

∑

n ωni ≤ 1 and

∑

n ǫni = 1

for each i. A similar relation can be set up for Assumption 2.

5.2.1 DAMA

Let us now apply above bounds to the DAMA result, based on data from a NaI detector.DAMA observes a relatively large unmodulated count rate of about 1 cnt/kg/day/keV com-pared to the modulation amplitude of about 0.02 cnt/kg/day/keV. In contrast, CoGeNTreports a modulation amplitude in the range of 0.5 to 1 cnt/kg/day/keV, compared to anunmodulated rate of 3 to 4 cnt/kg/day/keV. Because of this much smaller modulation am-plitude compared to the unmodulated rate for DAMA we expect the signal to be consistentwith our bounds. It is also well known that a consistent fit of the DAMA modulated and un-modulated data is possible e.g., for a Maxwellian halo, although in certain parameter regionsthe unmodulated rate does provide a non-trivial constraint, see e.g. [8, 11]. In figure 6 weshow the bounds for DAMA, assuming quenching factors of 0.3 for Na and 0.09 for I (con-stant in energy) using the spectral data on the modulated and unmodulated rates from [3].We find that both the general bound, eq. (5.11), as well as the bounds for a symmetrichalo, eq. (5.12), are consistent with the modulation even for a DM mass as small as 3GeV.As noted above the bounds get weaker for larger DM masses, hence they do not place anyrelevant constraint on the dark matter interpretation of the DAMA modulation signal.

In deriving the multi-target bounds eqs. (5.11) and (5.12) we had to use inequalitiesrelated to the presence of several contributions to the rates, which makes the bounds some-

– 17 –

JCAP03(2012)005

ææ

æ

ææ

ææ æ æ æ

ô

ô

ô

ô

ô

ô ô ô ô ô

ô

ôô

ôô

ô ô ô ô ô


DAMA: Na (Q = 0.3)


3 4 5 6

Eee HKeVL

Arb

itra

ryu

nit

s

æ æ æ æ æ æ æ æ æ æô

ôô ô ô

ô

ô

ô

ô

ô

ô ô ô ô ô ôô

ô

ô

ô

DAMA: I (Q = 0.09)



3 4 5 6

Eee HKeVL

Arb

itra

ryu

nit

s

Figure 7. Upper bound compared to the integrated modulation amplitude from DAMA data forAssumptions 2 and 2a, assuming that DM scatters only on sodium (left) or iodine (right). The reddots correspond to the l.h.s. of eq. (5.7) and the purple triangles to the r.h.s., for a DM mass ofmχ = 7GeV (left) and 30GeV (right). Bins are 0.5 keV wide and we sum all bins starting from bin ishown on the horizontal axis up to 7 keVee. Error bars are negligible and are not shown for clarity.

what weaker than single target bounds. In case of NaI we have a relatively large hierarchybetween the two targets since A = 23 for Na and A = 127 for I. Therefore, one may con-sider also the situation that only one of them dominates the signal. If scattering on bothtargets is kinematically allowed, iodine will dominate because of the A2 enhancement of thespin-independent scattering cross section (which we will assume here). For low DM masses,scattering on iodine may be kinematically forbidden since vm is larger than the escape veloc-ity of the halo and in that case the signal is provided only by scattering off Na (this requiresadditional assumptions on the escape velocity, which in general are not necessary for ourbounds). We show the bounds for Assumptions 2 and 2a assuming that scattering on eitherNa or I dominates in figure 7 for some representative DM masses. In both cases only veryweak constraints are obtained. While scattering on Na is consistent down to mχ ≈ 3GeVfor Assumption 2a, scattering on iodine is consistent for mχ & 30GeV under Assumption 2.(Note that below around 30GeV typically scattering on Na dominates.)

As expected, we conclude that based solely on our bounds the DAMA signal is com-patible with assuming dark matter scattering being its origin. We do not expect that thisconclusion will change significantly if bounds from procedure 2 according to eq. (5.13) wereapplied. Therefore, we limited ourselves to the discussion based on procedure 1 for DAMA.

6 Summary and discussion

The annual modulation is probably the most distinctive signature of dark matter and isplaying (and will certainly play) a central role in revealing its existence and nature. However,it is not always clear that the modulation detected by a DM direct detection experiment iscaused by the dark matter particles and not by any other unknown background source. Inthis work we have presented a consistency check for the amplitude of the modulation andthe unmodulated count rate, which is a necessary (but not sufficient) condition for DM beingthe origin of an observed annual modulation.

We have derived upper bounds on the energy integrated modulation amplitude in termsof the unmodulated rate by expanding the DM velocity integral to first order in the earth’svelocity ve ≈ 30 km/s. It holds for a wide class of particle physics models where the dif-

– 18 –

JCAP03(2012)005

ferential scattering cross section dσ/dEnr is proportional to 1/v2. Although we have onlyfocused on elastic scattering the method can also be generalized to the inelastic case. Theimportant aspect of our work is that our bounds hold for very general assumptions aboutthe DM velocity distribution f(v) in the sun’s vicinity. We have presented bounds under thehypothesis of a single DM species and the following assumptions on the DM halo:

• Assumption 1, bound in eq. (3.13): We assume that the only time dependence isinduced by the rotation of the earth around the sun. The halo itself is static on the timescale of months to years and spatially constant at the scale of the sun-earth distance.Otherwise the DM velocity distribution can have an arbitrary structure, including, forexample, several streams coming from various directions. In order to saturate thisbound the halo has to be very peculiar, with rather unrealistic properties.

• Assumption 2, bound in eq. (4.8): In addition to Assumption 1 we assume thatthere is just one preferred direction of the DM velocity distribution in the rest frame ofthe sun (independent of the minimal velocity vm in the halo integral). This requires cer-tain symmetries of the velocity distribution, which are specified in eqs. (4.3) and (4.4).It covers typically situations where the DM distribution is dominated by one singlecomponent, which may come from an arbitrary (but constant) direction. Assumption 2requires that the phase of the modulation is independent of the recoil energy (up to aphase flip of half a year).

• Assumption 2a, bound in eq. (4.9): We require that the preferred direction fromAssumption 2 is aligned with the motion of the sun. This is fulfilled for any isotropichalo, and also for tri-axial halos up to corrections due to the peculiar velocity of the sun.Furthermore, it includes the possibility of streams parallel to the motion of the sun,such as a possible dark disk. Assumption 2a requires that the maximum (or minimum)of the event rate is around June 2nd.

The theoretical bounds are obtained in terms of the minimal velocity vm and have tobe related to observable quantities by translating into recoil energy. We have outlined twopossible procedures for this task in section 5 for the case of elastic scattering, taking intoaccount statistical errors and the possibility that an unknown background may contribute tothe unmodulated rate, but that it has no time-dependence. As an example we have appliedthe proposed consistency checks to the annual modulation signals reported by the CoGeNTand DAMA experiments. While DAMA data are compatible with a dark matter origin for itsmodulation, severe restrictions on the dark matter mass can be set for the case of CoGeNT.Applying our bounds we find that the CoGeNT modulation amplitude can be consistentwith the unmodulated rate at the 68% CL only for DM masses mχ & 7.3, 18, 16GeV for theAssumptions 1, 2, 2a, respectively. If preliminary results on a possible surface events contami-nation of the unmodulated rate at low energies in CoGeNT are confirmed, those bounds wouldbecome even more restrictive. In this case, CoGeNT modulation data would be inconsistentwith the DM hypothesis under assumptions 2 and 2a at about 97% and 90% CL, respectively.DAMA has a relatively large unmodulated count rate of about 1 cnt/kg/day/keV comparedto the modulation amplitude of about 0.02 cnt/kg/day/keV. Therefore, our bounds are notvery stringent and the modulation amplitude is consistent with the unmodulated rate. Incontrast, CoGeNT reports a modulation amplitude in the range of 0.5 to 1 cnt/kg/day/keV,compared to an unmodulated rate of 3 to 4 cnt/kg/day/keV. Because of this relatively large

– 19 –

JCAP03(2012)005

ratio between modulated and unmodulated rate our method provides stringent constraintsin the case of CoGeNT. Several comments are in order:

(i) In deriving the bounds we assume a certain smoothness of the halo, since we are ex-panding in ve. Spikes in the velocity distribution much narrower than 30 km/s are notcovered by our procedure.

(ii) Our bounds assume that a modulation signal is present in the data. If the significanceof the modulation is weak, the bounds are more easily satisfied. This effect is explicitlyillustrated in section 5 by using CoGeNT data under different assumptions regarding themodulation signal. The method discussed here cannot be used to establish the presenceof a modulation, it can only test whether a given modulation signal is consistent withthe unmodulated rate.

(iii) In this work we have used the relation between the minimal velocity vm and nuclearrecoil energy Enr implied by elastic scattering, see eq. (2.2) and appendix B. The boundscan be generalised in a straightforward way also to inelastic scattering. However, inthat case vm = (mAEnr/µχA + δ)/

√2mAEnr has a minimum in Enr, and therefore

there is no one-to-one correspondence between vm and Enr. When translating fromvm to Enr one has to take into account that different disconnected regions in Enr cancontribute to a given interval in vm.

(iv) The type of DM-nucleus interaction (i.e., the particle physics) has to be specified beforeapplying our bounds. Apart from the 1/v2 dependence of the differential cross sectiondσ/dEnr, the Enr dependence of the interaction is encoded in the form factor F (Enr).It can describe conventional spin-independent or spin-dependent interactions, but alsoa possible non-trivial q2-dependence of the DM-nucleus interaction, all of which can beabsorbed into F (Enr).

To conclude, we have presented a consistency check for the amplitude of a DM inducedannual modulation compared to the unmodulated event rate in a given DM direct detectionexperiment. Our bounds rely only on very mild and realistic assumptions about the DM halo.We believe that the proposed method is a useful check which any annual modulation signalshould pass. A violation of our bounds suggests a non-DM origin of the annual modulation,or requires rather exotic properties of the DM distribution, for example very sharp spikes andedges. Such extreme features of the halo should have other observable consequences, suchas surprising spectral features or strong energy dependence of the modulation phase. Suchfeatures could be used as additional diagnostics, beyond the signatures explored here, whichare restricted to the energy integrated modulation amplitude, irrespective of the phase.

Acknowledgments

We thank Felix Kahlhofer for useful comments on the manuscript. T.S. and J.Z. are gratefulfor support from CERN for attending the CERN-TH institute DMUH’11, 18–29 July 2011,where this work was initiated. The work of T.S. was partly supported by the Transregio Son-derforschungsbereich TR27 “Neutrinos and Beyond” der Deutschen Forschungsgemeinschaft.J.H.-G. is supported by the MICINN under the FPU program and would like to thank thedivision of M. Lindner at MPIK for very kind hospitality.

– 20 –

JCAP03(2012)005

-0.1

0

0.1

rela

t. a

ccu

racy

0 200 400 600 800v

m [km/s]

10-7

10-6

10-5

10-4

10-3

η (unmodulated)

Aη (modul. ampl.)

solid: exact, dashed: approx.

unmod.

modulated

general bound

Figure 8. Comparison for the Maxwellian halo of the exact (solid line) and ve expanded (dashed)modulated and unmodulated halo integrals Aη and η, respectively. The modulation flips the phaseby half a year below the zero of the amplitude. In the lower panel the general bound (3.13) is alsoshown (dashed-dotted curve). The upper panel shows the relative accuracy defined as (exact —approx)/exact.

A Maxwellian halo

As a cross check we apply our formalism to the commonly used Maxwellian velocity distri-bution, with a cut off at an escape velocity vesc. In the galactic rest frame it is given by

fgal(v) ∝ [exp(−v2/v2)− exp(v2esc/v2)]Θ(vesc − v) , (A.1)

where we use v = 220 km s−1 and vesc = 550 km s−1. This distribution is boosted to the sun’sand earth’s rest frames as described in section 2. For the above velocity distribution the halointegral η(vm, t) is known analytically. We define the “exact” modulated and unmodulatedhalo integrals as

ηexact(vm) ≡ 1

2[η(vm, t∗) + η(vm, t∗ + 0.5)] ,

A′ηexact

(vm) ≡ 1

2[η(vm, t∗)− η(vm, t∗ + 0.5)] ,

(A.2)

where for the (exact) analytic expressions η(vm, t) we are are using the results given in theappendix of [27]. Above, t∗ is June 2nd and Aexact

η (vm) = |A′ηexact(vm)|.

We first check the accuracy of the ve expansion. Expanding to zeroth order in vegives the approximate unmodulated rate (3.3), while the modulation amplitude is given byexpanding to linear order in ve, (4.10), with sinαhalo = 0.5. The comparison with the “exact”

– 21 –

JCAP03(2012)005

expressions is shown in figure 8. Both for the modulation amplitude and the unmodulatedrate we find excellent agreement, with the differences hardly visible on logarithmic scale. Asseen from the upper panel, for large regions of vm the agreement is within few %, apart fromthe zero of the modulation amplitude. Minor deviations appear for large vm values, wherenon-linear effects due to the cut-off at the escape velocity become important. We see thatour expression for Aη captures accurately the phase flip of the modulation. At low vm theG(vm) term in eq. (4.10) dominates, leading to the maximum of the count rate in December,the modulation amplitude becomes zero when the two terms cancel, and the maximum inJune at large vm is obtained from the g(vm) term.

We compare next the bounds with the actual modulation amplitude. The dash-dotted(green) curve in figure 8 shows the general bound, eq. (3.13), which follows from Assump-tion 1. Clearly, the Maxwellian halo is far from saturating this bound. Much tighter con-straints are obtained from Assumptions 2 and 2a. Figure 9 shows the bounds (4.8) and (4.9)on integrals of modulation amplitudes. The shaded region shows the l.h.s. of the bounds,∫

vm1dvA′

η, where the upper boundary of the integration is chosen so high that the DM signalvanishes. Note that A′

η flips the sign at vm ∼ 200 km/s, which explains the maximum ofthe integral around 200 km/s. The curve labeled “partial bound” corresponds to the boundfollowing from Assumption 2, eq. (4.8), but without the second term in the square bracket.We find that it is only slightly more stringent than the general bound that follows fromAssumption 1, (3.13), especially for large vm. Although it does not capture the behaviour atlow vm it is rather similar to the full bound, eq. (4.8), (dashed curve) for vm & 400 km/s. Thebound from Assumption 2a, eq. (4.9) (dash-dotted curve), which assumes sinαhalo = 0.5, isroughly a factor 2 larger than the prediction for small vm1 and approaches the true modula-tion around vm1 & 400 km/s. In the right panel we show that for a Maxwellian distributionwith very small dispersion (v = 40 km/s) the bound for Assumption 2a is nearly saturated.As mentioned in section 3, the inequalities used to derive the bound (4.9) become equalities ifthe velocity distribution f(v) ∝ δ(ϑ), which is approximated by the Maxwellian with small v.

B Translating bounds in vm space to observable quantities

In sections 3 and 4 we have derived bounds involving the halo integral η(vm) and the modu-lation amplitude Aη(vm). Here we give details on how to translate these bounds into boundsthat involve observable rates as functions of nuclear recoil energy Enr. The nuclear energyis measured in electron-equivalents, Eee, which is related to Enr through a quenching factorQ. We also define a differential quenching factor q,

q ≡ dEee

dEnr= Q+ Enr

dQ

dEnr, where Q ≡ Eee

Enr, (B.1)

which then gives the translation between vm and Eee,

vm =

√

mAEee

2µ2χAQ

,1

ξ(Eee)≡ dvm

dEee=

dvmdEnr

dEnr

dEee=

1

2µχA

√

mAQ

2Eee

1

q. (B.2)

The quenching factor Q is in general energy dependent. If it is energy independent, thenq = Q, while if data are reported directly in Enr then furthermore q = Q = 1. Note thatq,Q, vm, ξ, are all positive.

– 22 –

JCAP03(2012)005

0 200 400 600 800v

m1 [km/s]

10-5

10-4

10-3

10-2

10-1

partial bound

symmetric halo

sym. halo, sinα = 0.5

integral of A’η

exact

v = 220 km/s

0 100 200 300 400v

m1 [km/s]

10-5

10-4

10-3

10-2

10-1

partial bound

symmetric halo

sym. halo, sinα = 0.5

integral of A’η

exact

v = 40 km/s

Figure 9. The bounds on the modulation amplitude for a Maxwellian velocity distribution andvelocity dispersions v = 220 km/s (40 km/s) in the left (right) panels. The shaded region shows theintegral of

∫

∞

vm1

A′

ηexact

. We show the bound for a “symmetric halo” from eq. (4.8), and the bound

from eq. (4.9) which assumes that the DM wind is parallel to the motion of the sun (“sym. halo,sinα = 0.5”). The solid curve (“partial bound”) shows the bound eq. (4.8), but without the secondterm in the square bracket.

The modulation amplitude and rate in units of counts/day/kg/keVee are given as

AR(Eee) = κAη(vm) , R(Eee) = κ η(vm) with κ(Eee) ≡CF 2(Eee/Q)

q(Eee), (B.3)

with the constant C defined in (2.6). To keep the notation simple we use the same symbolsfor rate and modulation whether they depend on recoil energy in keVee or keVnr. The unitsare denoted in the argument (Eee versus Enr) and should be clear from the context.

Let us consider first the general bound following from Assumption 1. Multiplyingeq. (3.13) by κ and converting the integral over vm into Eee by using eq. (B.2) one finds

AR(Eee) ≤ ve

−d(Rξ)

dEee+R

dξ

dEee+

√

2µ2χAQ

mAEee+

ξ

κ

dκ

dEee

− κ

∫

Eee

dEeeR

ξκ

2µ2χAQ

mAEee

(B.4)eq. (4.12) for Assumptions 2, 2a becomes similarly:

AR(Eee) ≤ ve sinαhalo

[

−d(Rξ)

dEee+R

(

dξ

dEee+

ξ

κ

dκ

dEee

)]

. (B.5)

Note that the constant C containing the cross section and local DM density drops out,as expected, since it is a common pre-factor for rate as well as modulation. In practise AR

and R are given in bins in Eee. Let us define the bin average of a quantity X(Eee) as

〈X〉i ≡1

∆Ei

∫ Ei2

Ei1

dEeeX(Eee) , (B.6)

– 23 –

JCAP03(2012)005

where Ei1 and Ei2 are the boundaries of bin i and ∆Ei = Ei2−Ei1. The observed modulationand rate in bin i are then Ai = 〈AR〉i and Ri =

⟨

R⟩

i, respectively. Now we have to perform

the bin average of eq. (B.4). The first term in the square bracket gives

R(Ei1)ξ(Ei1)−R(Ei2)ξ(Ei2)

∆Ei≈

Ri 〈ξ〉i −Ri+1 〈ξ〉i+1

∆Ei. (B.7)

For the other two terms in the square bracket we approximate 〈XY 〉 ≈ 〈X〉〈Y 〉 and theintegral becomes a sum over bins, up to the last reported bin i = N . Finally we obtain

Ai ≤ ve

Ri(αi + βi)−Ri+1α′i+1 − 〈κ〉i

N∑

j=i

Rjγj

, (B.8)

with

αi ≡〈ξ〉i∆Ei

, α′i ≡

〈ξ〉i∆Ei−1

, βi ≡⟨

dξ

dEee+

√

2µ2χAQ

mAEee+

ξ

κ

dκ

dEee

⟩

i

, γi ≡ ∆Ei

⟨

2µ2χAQ

mAEeeξκ

⟩

i

.

(B.9)For eq. (B.5) we get

Ai ≤ ve sinαhalo

[

Ri(αi + β′i)−Ri+1α

′i+1

]

with β′i ≡

⟨

dξ

dEee+

ξ

κ

dκ

dEee

⟩

i

. (B.10)

Terms with indices larger than N are dropped. We see from the definitions that αi, α′i, γi,

and 〈κ〉i are positive.A similar calculation for the bounds (4.8) and (4.9), following from the Assumptions 2

and 2a, leads toN∑

j=i

Ajxj ≤ ve sinα

Riyi − 〈vm〉iN∑

j=i+1

Rjγj

, (B.11)

with

xi ≡ ∆Ei

⟨

1

ξκ

⟩

i

, yi ≡⟨

1

κ

⟩

i

− 〈vm〉i γi , 〈vm〉i =⟨√

mAEee

2µ2χAQ

⟩

i

, (B.12)

and sinα = 1 (0.5) corresponds to Assumption 2 (2a).Any binning procedure will lead to inaccuracies. We have estimated binning effects by

slightly changing the prescription. Eq. (B.4) can be equivalently written as

AR(Eee) ≤ ve

−ξdR

dEee+R

√

2µ2χAQ

mAEee+

ξ

κ

dκ

dEee

− κ

∫

Eee

dEeeR

ξκ

2µ2χAQ

mAEee

. (B.13)

The first term in the square bracket can be estimated as

− ξdR

dEee≈ 〈ξ〉i

Ri −Ri+1

∆Ei, (B.14)

and the other terms are bin-averaged similar as above. We have calculated the CoGeNTbounds also using this binning procedure and obtain similar results as with our defaultmethod. This confirms that inaccuracies due to binning are acceptable for present Co-GeNT data.

– 24 –

JCAP03(2012)005

unmod. rate from [4] corrected rate [21]

Mean 68% 95% Mean 68% 95%

Ass. 1: general bound 10 5 − 12.5 5 −Ass. 2: symmetric halo 29.5 17 7 63 34.5 16

Ass. 2a: sym. halo, sinα = 0.5 37.5 14 − 94.5 30.5 −

Table 3. Lower bounds on the DM mass in GeV, from the requirement that the CoGeNT modu-lation amplitude is consistent with the upper bound from the unmodulated rate, according to theAssumptions 1, 2, 2a on the DM distribution obtained as described in appendix C. In the left partof the table we use the published unmodulated event rates from [4], whereas for the right part of thetable we adopt the preliminary results on surface events contamination at low energies from [21].

C Alternative procedure for optimizing the bound in presence of unknown

background

In section 5 we have outlined a method based on a least-square minimization (“procedure 2”)to find “optimal” values for the coefficients ωi parameterizing the background contributionin each bin. Here we provide an alternative method for this task.

Let us consider the inequalities (5.4) and (5.5) as a system of equations for the ωi. Weobserve that they have a triangular structure with respect to ωi: the inequality for a givenindex i depends only on ωj with j ≥ i, i.e., the inequality for i = N depends only on ωN ,for i = N − 1 it depends on ωN−1, ωN , and so on. From the fact that αi, 〈κ〉i , γi are positivefollows that the bound for a given i is weakest if ωi is as large as possible and ωj with j > iare as small as possible. We can implement this requirement in an iterative way: replacingthe inequality sign by an equality, we obtain a system of N linear equations in ωi. We cansolve this system by starting at i = N and going up step by step to i = 1, in each stepobtaining a value for the corresponding ωi, to be used in the next step. If the solution for ωi

is less than one, it will be the smallest allowed value, and hence the bound for i− 1 will beweakest. If the solution for ωi is larger than one the bound is violated by bin i and we haveto set the corresponding ωi = 1 (the largest physically allowed value). In that way we obtaina set of ωi corresponding to the most conservative choice of background contributions to theunmodulated rate.

This method treats the inequalities as “hard bounds” and does not take into accountthe fact that the amplitudes are subject to experimental uncertainties. A conservative wayto include uncertainties is to apply this procedure as described above but using instead ofthe l.h.s. of (5.4) and (5.5) its central value minus n standard deviations. In this way oneaims to satisfy the inequalities within the nσ interval for each bin. We show the results ofsuch an analysis for CoGeNT data in table 3. Typically one finds qualitatively similar butslightly stronger bounds than for procedure 1, although if the method is considered at higherCL bounds may become weaker or even disappear, as visible in the table for the columns at95% CL. Note, however, that subtracting nσ from the mean value for all bins leads to veryconservative limits.

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– 25 –

JCAP03(2012)005

[2] K. Freese, J.A. Frieman and A. Gould, Signal Modulation in Cold Dark Matter Detection,Phys. Rev. D 37 (1988) 3388 [INSPIRE].

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(2011) 103514 [arXiv:1011.1915] [INSPIRE].

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– 27 –

Astrophysics-Independent Bounds on the Annual Modulation of Dark Matter Signals

Juan Herrero-Garcia,1,* Thomas Schwetz,2,† and Jure Zupan3,‡

1Departamento de Fisica Teorica and IFIC, Universidad de Valencia CSIC, Paterna, Apartado 22085, E-46071 Valencia, Spain2Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

3Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA(Received 8 May 2012; published 4 October 2012)

We show how constraints on the time integrated event rate from a given dark matter (DM) direct

detection experiment can be used to bound the amplitude of the annual modulation signal in another

experiment. The method requires only mild assumptions about the properties of the local DM distribution:

that it is temporally stable on the scale of months and spatially homogeneous on the ecliptic. We apply the

method to the annual modulation signal in DAMA/LIBRA, which we compare to the bounds derived from

XENON10, XENON100, cryogenic DM search, and SIMPLE data. Assuming a DM mass of 10 GeV, we

show that under the above assumptions about the DM halo, a DM interpretation of the DAMA/LIBRA

signal is excluded for several classes of models: at 6:3! (4:6!) for elastic isospin conserving (violating)

spin-independent interactions, and at 4:9! for elastic spin-dependent interactions on protons.

DOI: 10.1103/PhysRevLett.109.141301 PACS numbers: 95.35.+d

Dark matter (DM) constitutes a significant fraction ofthe energy density in the Universe, !DM ¼ 0:229" 0:015[1]. This conclusion is based entirely on gravitationaleffects of DM. A fundamental question is whether DMinteracts also nongravitationally. Direct detection experi-ments, for instance, are looking for the scattering of DMparticles from the galactic halo in underground detectors.A characteristic feature of the signal is an annual modu-lation, because Earth rotates around the Sun, while at thesame time the Sun moves relative to the DM halo [2].At present, two experiments are reporting annually modu-lated signals, DAMA/LIBRA [3] (DAMA for short) andCoGeNT [4], with significances of 8:9! and 2:8!, res-pectively. Assuming the standard Maxwellian DM haloand elastic spin-independent (SI) DM scattering, bothclaims are in tension [5,6] with bounds on time integratedrates from other experiments such as XENON10 [7],XENON100 [8], or cryogenic DM search (CDMS) [9].The situation may change in the case of non-MaxwellianDM halos, e.g., halos with anisotropic velocity distri-butions or with significant substructure, for instance DMstreams or DM debris flows. Recently CDMS provided adirect bound on the modulation signal, which disfavorsthe CoGeNT modulation without referring to any halo orparticle physics model [10]. Therefore, we focus below onDAMA and CoGeNT.

In this Letter, we present a general method that avoidsastrophysical uncertainties when comparing putative DMmodulation signals with bounds on time averaged DMscattering rates from different experiments by combiningthe results from Refs. [11,12] with bounds on the modu-lation derived by us in Ref. [13]. We are then able totranslate the bound on the DM scattering rate in oneexperiment into a bound on the annual modulationamplitude in a different experiment. The resulting

bounds present roughly an order of magnitude improve-ment over Refs. [11–13].The bounds are (almost completely) astrophysics inde-

pendent. Only very mild assumptions about DM haloproperties are used: (i) that it does not change on thetime scales of months, (ii) that the density of DM in thehalo is constant on the scales of the Earth-Sun distance, and(iii) that the DM velocity distribution does not varystrongly on scales of the Earth velocity, ve ¼ 29:8 km=s.If the modulation signal is due to DM, then the modulationamplitude has to obey the bounds. In the derivation, anexpansion in ve over the typical DM velocity #200 km=sis used. The validity of the expansion can be checkedexperimentally, by searching for the presence of higherharmonics in the time-stamped DM scattering data [13].Bounds on the annual modulation.—We focus on elastic

scattering of DM " off a nucleus (A, Z), depositing thenuclear-recoil energy Enr. The differential rate in events/keV/kg/day is

RAðEnr; tÞ ¼#"!

0A

2m"$2"A

F2AðEnrÞ%ðvm; tÞ; (1)

with #" the local DM density, !0A the total DM-nucleus

scattering cross section at zero momentum transfer,m" the

DMmass,$"A the DM-nucleus reduced mass, and FAðEnrÞa nuclear form factor. For SI interactions,!0

A can bewritten

as!SIA ¼ !p½Zþ ðA( ZÞðfn=fpÞ)2$2

"A=$2"p, where!p is

the DM-proton cross section and fn;p are coupling

strengths to neutron and proton, respectively. Apart from#", the astrophysics enters in Eq. (1) through the halo

integral

%ðvm; tÞ *Zv>vm

d3vfdetðv; tÞ

v; vm ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffimAEnr

2$2"A

s; (2)

PRL 109, 141301 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

5 OCTOBER 2012

0031-9007=12=109(14)=141301(5) 141301-1 ! 2012 American Physical Society

where vm is the minimal velocity required for recoil energyEnr, and fdetðv; tÞ describes the distribution of DM particlevelocities in the detector rest frame with fdetðv; tÞ # 0 andRd3vfdetðv; tÞ ¼ 1. The integral of the velocity distribu-

tion enters in RAðEnr; tÞ because DM scattering at differentangles probes different DM velocities even for fixed Enr.The velocity distributions in the rest frames of the detectorand the Sun are related by fdetðv; tÞ ¼ fsun½vþ veðtÞ',where veðtÞ is the velocity vector of Earth. The rotationof the Earth around the Sun introduces a time dependencein the event rate through !ðvm; tÞ ¼ !!ðvmÞ þ "!ðvm; tÞ,where

"!ðvm; tÞ ¼ A!ðvmÞ cos2#½t( t0ðEnrÞ'; (3)

when expanding to first order in ve ¼ 29:8 km=s )vsun ’ 230 km=s, and A!ðvmÞ # 0. The expansion can be

truncated, if fsunðvÞ does not show large variations on thescale of ve, i.e., vejdfðvÞ=dvj ) fðvÞ. We are also assum-ing that the only time dependence comes from the rotationof Earth around the Sun and that fsunðvÞ itself is constant intime and space.

Under those assumptions we schematically have !!*RfðvÞ=v and A! * ve

RfðvÞ=v2. After some algebra, the

modulation amplitude A!ðvmÞ can be shown to be boundedby the unmodulated halo integral !! (see Ref. [13] for thederivation):

A!ðvmÞ + ve

!( d !!

dvm

þ !!ðvmÞvm

(Zvm

dv!!ðvÞv2

": (4)

If we further assume that the DM velocity distributionobeys certain symmetry properties such that there is onlyone single direction related to the DM flow, independent ofvm (see Ref. [13] for details), then one obtains a morestringent constraint:

Z v2

v1

dvmA!ðvmÞ + sin$ve

!!!ðv1Þ ( v1

Zv1

dv!!ðvÞv2

":

(5)

Here $ is the angle between the DM flow and the directionorthogonal to the ecliptic. The most conservative bound isobtained for sin$ ¼ 1 (which would correspond to a DMstream parallel to the ecliptic). However, in many cases theDM flow will be aligned with the motion of the Sun withinthe Galaxy. This holds for any isotropic velocity distribu-tion and, up to a small correction due to the peculiarvelocity of the Sun, also for triaxial halos or a significantcontribution from a possible dark disk. In this case, wehave sin$ ’ 0:5.

In the following, we will use time averaged rates fromvarious experiments to derive an upper bound on !!ðvmÞ.In order to be able to apply this information we integrateEq. (4) over vm and drop the negative terms in Eqs. (4)and (5). This gives the bounds

Z v2

v1

dvmA!ðvmÞ + ve

!!!ðv1Þ þ

Z v2

v1

dv!!ðvÞv

"; (6)

Z v2

v1

dvmA!ðvmÞ + sin$ve !!ðv1Þ: (7)

In practice, the integrals on the left-hand side (LHS) arereplaced by a sum over bins. Below we will refer to therelations (6) and (7) with sin$ ¼ 0:5 as the bounds forgeneral halo and symmetric halo, respectively. Heresymmetric refers to the local velocity distribution accord-ing to the sentence before Eq. (5), not the spatial distri-bution of DM.Bounds on the unmodulated halo integral.—Let us first

consider SI scattering with fn ¼ fp. Generalization to

isospin-violating scattering with fn ! fp and to spin-

dependent (SD) scattering is straightforward. The pre-dicted number of events in an interval of observed energies[E1, E2] is

Npred½E1;E2' ¼ MTA2

Z 1

0dEnrF

2AðEnrÞG½E1;E2'ðEnrÞ~!ðvmÞ:

(8)

Here G½E1;E2'ðEnrÞ is the detector response function, whichdescribes the contribution of events with true nuclear-recoilenergy Enr to the observed energy interval [E1, E2]. It maybe nonzero outside the Enr 2 ½E1; E2' interval due to thefinite energy resolution and includes also (possibly energydependent) efficiencies.M and T are the detector mass andexposure time, respectively, and

~! , %p&'

2m'(2'p

!!; (9)

where ~! has units of events/kg/day/keV.Now we can use the fact that ~! is a monotonically

decreasing function of vm [11] (see also Refs. [14,15]).Among all possible forms for ~! such that they pass through~!ðvmÞ at vm, the minimal number of events is obtainedfor ~! constant and equal to ~!ðvmÞ until vm and zero after-wards. Therefore, for a given vm we have a lower bound

Npred½E1;E2'ðvmÞ # (ðvmÞ with

(ðvmÞ ¼ MTA2 ~!ðvmÞZ EðvmÞ

0dEnrF

2AðEnrÞG½E1;E2'ðEnrÞ;

(10)

where EðvmÞ is given in (2). Suppose an experiment ob-serves Nobs

½E1;E2' events in the interval [E1, E2]. Then we can

obtain an upper bound on ~! for a fixed vm at a confidencelevel (C.L.) by requiring that the probability of obtainingNobs

½E1;E2' events or less for a Poisson mean of(ðvmÞ is equalto 1-C.L. Note that this is actually a lower bound on theC.L., because Eq. (10) provides only a lower bound onthe true Poisson mean. For the same reason, we cannot usethe commonly applied maximum-gap method to derive a


5 OCTOBER 2012

141301-2

bound on ~!. If several different nuclei are present, therewill be a corresponding sum in Eqs. (8) and (10).

The limit on ~! can then be used in the right-hand side(RHS) of Eq. (6) or (7) to constrain the modulation ampli-tude. For concreteness we first focus on the annual modu-lation in DAMA. If m" is around 10 GeV, then DM

particles do not have enough energy to produce iodinerecoils above the DAMA threshold. We can thus assumethat the DAMA signal is entirely due to the scattering on

sodium. We define ~A! ! #p$"=ð2m"%2"pÞA!, which is

related to the observed modulation amplitude Aobsi by

~A obs! ðvi

mÞ ¼Aobsi qNa

A2NahF2

NaiifNa: (11)

Here qNa ¼ dEee=dEnr is the sodium quenching factor, forwhich we take qNa ¼ 0:3. The index i labels energy bins,with vi

m given by the corresponding energy bin centerusing Eq. (2). Further, hF2

Naii is the sodium form factoraveraged over the bin width and fNa ¼ mNa=ðmNa þmIÞ isthe sodium mass fraction. For the modulation amplitude inCoGeNTwe proceed analogously. Note that the conversion

factor from !! to ~! is the same as for A! to ~A!, and is not

dependent on the nucleus. Therefore, the bounds (6) and

(7) apply to ~!, ~A! without change, even if the LHS and

RHS refer to different experiments.Let us briefly describe the data we use to derive the

upper bounds on ~!. We consider results from XENON10[7] (XE10) and XENON100 [8] (XE100). In both caseswe take into account the energy resolution due to Poissonfluctuations of single electrons. For XE100, we adopt thebest-fit light-yield efficiency Leff from Ref. [8]. The XE10analysis is based on the so-called S2 ionization signal andwe follow [7] imposing a sharp cut off of the efficiencybelow threshold. From CDMS, we use results from a low-threshold (LT) analysis [9] of Ge data, as well as data on Si[16]. For SIMPLE [17], we use the observed number ofevents and expected background events to calculate thecombined Poisson probability for stages 1 and 2.

For all experiments we use the lower bound on theexpected events, Eq. (10), to calculate the probability ofobtaining less or equal events than observed. For XE100,CDMS Si, and SIMPLE we just use the total number ofevents in the entire reported energy range. For XE10 andCDMS LT, the limit can be improved if data are binned andthe probabilities for each bin are multiplied. This assumesthat the bins are statistically independent, which requires tomake bins larger than the energy resolution. For XE10 weonly use two bins. For CDMS LT, we combine the 36 binsfrom Fig. 1 of Ref. [9] into 9 bins of 2 keV where theenergy resolution is 0.2 keV.

Results.—In Fig. 1 we show the 3# limits on ~! com-

pared to the modulation amplitudes ~A! from DAMA and

CoGeNT for a DM mass of 10 GeV. Similar results havebeen presented in Refs. [14,15]. The CoGeNT amplitude

depends on whether the phase is let to vary freely in the fitor fixed at June 2nd [6], which applies to the general andsymmetric halos, respectively. Already at this level XE100is in tension with the modulation from DAMA (and tosome extent also CoGeNT).We now apply our method. As shown in Fig. 2, the null

results become significantly more constraining after apply-ing the bound Eq. (6). DAMA and CoGeNT are stronglyexcluded by XE100, XE10, CDMS LT already for thegeneral halo, and even more assuming a symmetric halo.Then also CDMS Si excludes DAMA, and there is sometension with SIMPLE (data not shown). In Fig. 3, weconsider two variations of DM-nucleus interaction. Theupper panel is for SD interactions with the proton, wherethe bound from SIMPLE is in strong disagreement with theDAMA modulation, due to the presence of fluorine intheir target. (A comparable limit has been published re-cently by PICASSO [18].) In the lower panel of Fig. 3 weshow the case of SI isospin-violating interactions with

FIG. 1 (color online). 3# upper bounds on ~!. The modulationamplitude ~A! is shown for DAMA (for qNa ¼ 0:3) and CoGeNT

for both free phase fit (general) and fixing the phase to June 2nd(symmetric). We assume a DM mass of 10 GeV and SIinteractions.

FIG. 2 (color online). Integrated modulation signals,Rv2v1dvA~!, from DAMA and CoGeNT compared to the 3# upper

bounds for the general halo, Eq. (6). We assume SI interactionsand a DM mass of 10 GeV. The integral runs from v1 ¼ vmin tov2 ¼ 743 km=s (end of the 12th bin in DAMA).


5 OCTOBER 2012

141301-3

fn=fp ¼ "0:7. This choice evades bounds from Xe, but

now the DAMA modulation is excluded by the boundsfrom CDMS Si for the general halo and CDMS Si, LT,and SIMPLE for the symmetric halo.

Let us now quantify the disagreement between theobserved DAMA modulation and the rate from anothernull-result experiment using our bounds. We first fix vm.To each value of ~!ðvmÞ, Eq. (10) provides a Poissonmean "ðvmÞ. We can then calculate the probability p! to

obtain equal or less events than measured by the null-result experiment. Then we construct the bound on themodulation using the same value ~!ðvmÞ on the RHS ofEq. (6) or (7) [the integrand ~!ðvÞ in Eq. (6) is calculatedusing the same p! but with v > vm in Eq. (10)]. We

calculate the probability pA that the bound is not vio-lated by assuming on the LHS of Eq. (6) or (7) aGaussian distribution for the DAMA modulation signalwith the measured standard deviations in each bin. Thenpjointð~!Þ ¼ p!pA is the combined probability of obtain-

ing the experimental result for the chosen value of ~!.Then we maximize pjointð~!Þ with respect to ~! to obtain

the highest possible joint probability.The results of such an analysis are shown in Fig. 4.

The analysis is performed at the fixed vm correspondingto the third modulation data point in DAMA, dependingon the DM mass m#. We find that for all considered

interaction types and m# & 15 GeV at least one experi-

ment disfavors a DM interpretation of the DAMA modu-lation at more than 4$ even under the very modestassumptions of the general halo. In the case of SI inter-actions, the tension with XE100 is at more than 6$ form# * 8 GeV and saturates at the significance of the modu-

lation data point itself at about 6:4$ for m# * 13 GeV.

The exclusion from XE10 is nearly independent of the DMmass slightly below 6$. We show also a few examples ofthe joint probability in case of a symmetric halo.As mentioned above, one requirement for our method to

apply is that the DM velocity distribution fðvÞ is smoothon scales & ve. Results from N-body simulations [19]indicate that close to the galactic escape velocity vesc %550 km=s fluctuations at such small scales can becomesignificant due to the presence of cold unvirialized DMstreams. Note however, that in all cases shown above theDAMA modulation signal is excluded already for vmin

well below vesc, where fðvÞ is expected to be sufficientlysmooth [19]. Furthermore, because Oðv2

eÞ terms in theexpansion of Eq. (1) lead to the appearance of a ½cosð2%tÞ'2time dependence, the validity of this approximation can bechecked experimentally by searching for higher harmonics inthe modulation.While astrophysics uncertainties are avoided, the ob-

tained bounds are still subject to nuclear, particle physics,and experimental uncertainties. For instance, the tensionbetween the DAMA signal and the bounds depends on thevalue of the Na quenching factor qNa, light-yield or ion-ization yield efficiencies in Xe, upward fluctuations frombelow threshold, and so on. For example, if a value ofqNa ¼ 0:45 is adopted instead of the fiducial value of 0.3consistency for SD and isospin-violating interactions can

FIG. 3 (color online). Integrated modulation signalRv2vmin

dvA~!

from DAMA compared to the 3$ upper bounds for the generalhalo, Eq. (6) (solid), and symmetric halo, Eq. (7) with sin& ¼0:5 (dotted). We assume a DM mass of 10 GeV, and SDinteractions on protons (upper panel) and SI interactions withfn=fp ¼ "0:7 (lower panel).

5 10 15 20mχ (GeV)

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Prob

abili

ty

3

4

5

6

7

Stan

dard

dev

iatio

ns

SIMPLE (SD)SIMPLE (IV)

CDMS LT (IV)

CDMS Si (IV) XE100 (SI)

XE10 (SI)

CDMS LT (SI)

CDMS Si (IV)

XE10 (SI)

solid: general halo, dashed: symmetric halo

FIG. 4 (color online). The probability that the integratedmodulation amplitude in DAMA (summed starting from thethird bin) is compatible with the bound derived from the con-straints on ~! for various experiments as a function of the DMmass. The label SI (SD), refers to spin-independent (spin depen-dent) interactions with fn ¼ fp (fn ¼ 0), and IV refers to

isospin-violating SI interactions with fn=fp ¼ "0:7. For solid

and dashed curves we use the bounds from Eqs. (6) and (7),respectively.


5 OCTOBER 2012

141301-4

be achieved in the case of the general halo at around 3!,while for SI interactions the XE10 bound still impliestension at more than 5! for m" * 10 GeV. Hence, the

precise C.L. of exclusion may depend on systematic un-certainties. We also stress that the above bounds apply toelastic scattering only.

In conclusion, we have presented a powerful method tocheck the consistency of an annual modulation signal in aDM direct detection experiment with bounds on the totalDM scattering rate from other experiments, almost com-pletely independent of astrophysics, for a given type ofDM-nucleus interaction. While our bounds strongly dis-favor a DM interpretation of present annually modulatedsignals for several models of DM interactions (SI and SDelastic scattering), the method will be an important test thatany future modulated signal will have to pass before a DMinterpretation can be accepted.

J. H.-G. is supported by the MICINN under the FPUprogram. J. H.-G. and T. S. acknowledge support fromthe EU FP7 ITN INVISIBLES (MC Actions, GrantNo. PITN-GA-2011-289442). J. Z. was supported in partby the U. S. National Science Foundation under GrantNo. PHY1151392.

*juan.a.herrero AT uv.es†schwetz AT mpi-hd.mpg.de‡zupanje AT ucmail.uc.edu

[1] E. Komatsu et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 192, 18 (2011).

[2] A. K. Drukier, K. Freese, and D.N. Spergel, Phys. Rev. D33, 3495 (1986); K. Freese, J. A. Frieman, and A. Gould,Phys. Rev. D 37, 3388 (1988).

[3] R. Bernabei et al. (DAMA Collaboration), Eur. Phys. J. C56, 333 (2008); , 67, 39 (2010).

[4] C. E. Aalseth et al. (CoGeNT Collaboration), Phys. Rev.Lett. 107, 141301 (2011).

[5] T. Schwetz and J. Zupan, J. Cosmol. Astropart. Phys. 08(2011) 008; J. Kopp, T. Schwetz, and J. Zupan, J. Cosmol.Astropart. Phys. 03 (2012) 001; M. Farina, D.Pappadopulo, A. Strumia, and T. Volansky, J. Cosmol.Astropart. Phys. 11 (2011) 010.

[6] P. J. Fox, J. Kopp, M. Lisanti, and N. Weiner, Phys. Rev. D85, 036008 (2012).

[7] J. Angle et al. (XENON10 Collaboration), Phys. Rev. Lett.107, 051301 (2011).

[8] E. Aprile et al. (XENON100 Collaboration), Phys. Rev.Lett. 107, 131302 (2011).

[9] Z. Ahmed et al. (CDMS-II Collaboration), Phys. Rev.Lett. 106, 131302 (2011).

[10] Z. Ahmed et al. (CDMS Collaboration), arXiv:1203.1309.[11] P. J. Fox, J. Liu, and N. Weiner, Phys. Rev. D 83, 103514

(2011).[12] P. J. Fox, G. D. Kribs, and T.M. P. Tait, Phys. Rev. D 83,

034007 (2011).[13] J. Herrero-Garcia, T. Schwetz, and J. Zupan, J. Cosmol.

Astropart. Phys. 03 (2012) 005.[14] M. T. Frandsen, F. Kahlhoefer, C. McCabe, S. Sarkar, and

K. Schmidt-Hoberg, J. Cosmol. Astropart. Phys. 01(2012) 024.

[15] P. Gondolo and G. B. Gelmini, arXiv:1202.6359.[16] D. S. Akerib et al. (CDMS Collaboration), Phys. Rev. Lett.

96, 011302 (2006).[17] M. Felizardo et al. (SIMPLE Collaboration), Phys. Rev.

Lett. 108, 201302 (2012).[18] S. Archambault et al. (PICASSO Collaboration), Phys.

Lett. B 711, 153 (2012).[19] M. Kuhlen, N. Weiner, J. Diemand, P. Madau, B. Moore,

D. Potter, J. Stadel, and M. Zemp, J. Cosmol. Astropart.Phys. 02 (2010) 030.


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JCAP07(2013)049

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Halo-independent methods forinelastic dark matter scattering

Nassim Bozorgnia,a Juan Herrero-Garcia,b Thomas Schwetza

and Jure Zupanc

aMax-Planck-Institut fur Kernphysik,Saupfercheckweg 1, 69117 Heidelberg, GermanybDepartamento de Fisica Teorica, and IFIC, Universidad de Valencia-CSIC,Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, SpaincDepartment of Physics, University of Cincinnati,Cincinnati, Ohio 45221, U.S.A.

E-mail: [email protected], [email protected],[email protected], [email protected]

Received May 21, 2013Accepted July 6, 2013Published July 31, 2013

Abstract. We present halo-independent methods to analyze the results of dark matter directdetection experiments assuming inelastic scattering. We focus on the annual modulationsignal reported by DAMA/LIBRA and present three different halo-independent tests. First,we compare it to the upper limit on the unmodulated rate from XENON100 using (a) thetrivial requirement that the amplitude of the annual modulation has to be smaller thanthe bound on the unmodulated rate, and (b) a bound on the annual modulation amplitudebased on an expansion in the Earth’s velocity. The third test uses the special predictionsof the signal shape for inelastic scattering and allows for an internal consistency check ofthe data without referring to any astrophysics. We conclude that a strong conflict betweenDAMA/LIBRA and XENON100 in the framework of spin-independent inelastic scatteringcan be established independently of the local properties of the dark matter halo.

Keywords: dark matter theory, dark matter experiments


c© 2013 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2013/07/049

JCAP07(2013)049

Contents

1 Introduction 1

2 Notation 2

3 Bound on the annual modulation amplitude from the expansion in ve 4

4 Halo-independent tests for inelastic scattering 6

5 Conclusions 11

1 Introduction

If dark matter (DM) is a “Weakly Interacting Massive Particle” (WIMP) it may induce anobservable signal in underground detectors by depositing a tiny amount of energy after scat-tering with a nucleus in the detector material [1]. Many experiments are currently exploringthis possibility and delivering a wealth of data. Among them is the DAMA/LIBRA exper-iment [2] (DAMA for short) which reports the striking signature of an annual modulationof the signal in their NaI scintillator detector, with a period of one year and a maximumaround June 2nd with very high statistical significance. Such an effect is expected for DMinduced events because the velocity of the detector relative to the DM halo changes due tothe Earth’s rotation around the Sun [3, 4].

Assuming elastic spin-independent interactions the DAMA modulation signal is instrong tension with constraints on the total DM interaction rate from other experiments [5–8]. This problem can be alleviated by considering inelastic scattering [9], where the DMparticle χ up-scatters to an excited state χ∗ with a mass difference δ = mχ∗ −mχ compa-rable to the kinetic energy of the incoming particle, which is typically O(100) keV. Underthis hypothesis scattering off the heavy iodine nucleus is favoured compared to the relativelylight sodium in the NaI crystal used in DAMA. Furthermore, the relative strength of themodulation signal compared to the unmodulated rate can be enhanced.1 Nevertheless, underspecific assumptions for the DM halo — typically a Maxwellian velocity distribution — alsothe inelastic scattering explanation of the DAMA signal is in tension with the bounds fromXENON100 [12] and CRESST-II [10], see e.g., [13–16]. Below we show that this conclusioncan be confirmed in a halo-independent way. Let us mention that any explanation of theDAMA signal based on iodine scattering is disfavoured also by KIMS results [17], since their90% CL upper bound on the DM scattering rate on CsI is already somewhat lower than thesize of the modulation amplitude observed in DAMA. This tension is completely independentof astrophysics as well as particle physics as long as scattering happens on iodine.

For typical inelastic scattering explanations of DAMA the mass splitting between thetwo DM states is chosen such that the minimal velocity, vm, required to deposit the thresholdenergy in the detector is already close to the galactic escape velocity, vesc. Only DM particles

1The possibility to use inelastic scattering to reconcile the event excess observed in CRESST-II [10] withother bounds has been discussed in [11]. Here we do not follow this hypothesis and focus on the DAMAmodulation signal.

– 1 –

JCAP07(2013)049

with velocities in the interval, v ∈ [vesc − ∆v, vesc], contribute, where ∆v is the range ofminimal velocities probed by the experiment and is comparable to the Earth’s velocity aroundthe Sun, ve ≈ 30 km/s. In this case the DM direct detection experiment probes the tails ofthe DM velocity distribution, where halo-substructures such as streams or debris flows areexpected. The results are thus quite sensitive to the exact history of the Milky Way halo,mergers, etc, and significantly depend on the halo properties, see e.g. [18, 19]. Thereforeit is important to develop astrophysics-independent methods to evaluate whether the aboveconclusion on the disagreement of the DAMA signal with other bounds is robust with respectto variations of DM halo properties.

An interesting method to compare signals and/or bounds from different experimentsin an astrophysics independent way has been proposed in refs. [20, 21]. This so-called vm-method has been applied in various recent studies for elastic scattering, see e.g., [22–27]. Thegeneralization of this method to inelastic scattering involves some complications which weare going to address in detail below.

In part of our analyses we will also use the fact that ve is small compared to all othertypical velocities in the problem. One can then derive astrophysics-independent bounds onthe annual modulation signal [28] by expanding in ve and relating the O(v0

e) and O(ve) termsin a halo-independent way. In [25, 28] the expansion was applied to the case of elastic DMscattering with DM masses of order 10 GeV, where the expansion is expected to be well-behaved, and it has been shown that for elastic scattering a strong tension between DAMAand constraints from other experiments can be established independent of the details of theDM halo. In the following we will generalize this type of analysis to the case of inelasticscattering, where special care has to be taken about whether the expansion in ve remainswell-behaved.

Below we will present three different tests for the consistency of the inelastic scat-tering interpretation of the DAMA signal, focusing on the tension with the bound fromXENON100 [8]:

• the “trivial bound” obtained by the requirement that the amplitude of the annualmodulation has to be smaller than the unmodulated rate,

• the bound on the annual modulation signal based on the expansion of the halo integralin ve, and

• a test based on the predicted shape of the signal in the case of inelastic scattering whichwe call the “shape test” in the following.

The paper is structured as follows. We fix basic notation in section 2. In section 3we discuss the bound on the annual modulation amplitude derived in [28]. By identifyingthe relevant expansion parameter we point out its limitations in the case of inelastic scat-tering. In section 4 we develop halo-independent methods for inelastic scattering, focusingon the tension between DAMA and XENON100, and apply the three different types of testsmentioned above. Conclusions are presented in section 5.

2 Notation

The differential rate in events/keV/kg/day for DM χ to scatter off a nucleus (A,Z) anddeposit the nuclear recoil energy Enr in the detector is

R(Enr, t) =ρχmχ

1

mA

∫v>vm

d3vdσAdEnr

vfdet(v, t) . (2.1)

– 2 –

JCAP07(2013)049

Emin E2E1

umin

umax

Emed

umed

Enr

v m

Figure 1. vm as a function of Enr for the case of inelastic scattering for some arbitrary δ > 0.

Here ρχ ' 0.3 GeV/cm3 is the local DM density, mA and mχ are the nucleus and DM masses,σA the DM-nucleus scattering cross section and v the 3-vector relative velocity between DMand the nucleus, while v ≡ |v|. For a DM particle to deposit a recoil energy Enr in thedetector, a minimal velocity vm is required, restricting the integral over velocities in eq. (2.1).For inelastic scattering we have

vm =

√1

2mAEnr

(mAEnr

µχA+ δ

), (2.2)

where µχA is the reduced mass of the DM-nucleus system, and δ is the mass splitting betweenthe two dark matter states. Note that for each value of Enr there is a corresponding vm whilethe converse is not always true. Certain values of vm correspond to two values of Enr, othersmaybe to none. This is illustrated in figure 1, where we plot vm as a function of Enr for somearbitrary δ > 0.

The particle physics enters in eq. (2.1) through the differential cross section. For thestandard spin-independent and spin-dependent scattering the differential cross section is

dσAdEnr

=mA

2µ2χAv

2σ0AF

2(Enr) , (2.3)

where σ0A is the total DM-nucleus scattering cross section at zero momentum transfer, and

F (Enr) is a form factor. We focus here on spin-independent inelastic scattering. We alsoassume that DM couples with the same strength to protons and neutrons (fp = fn). Relaxingthis assumption does not change the conclusions since DM particles scattering on Xe and Ihave very similar dependence on fn/fp (cf. figure 5 of [11]). The astrophysics dependenceenters in eq. (2.1) through the DM velocity distribution fdet(v, t) in the detector rest frame.Defining the halo integral

η(vm, t) ≡∫v>vm

d3vfdet(v, t)

v, (2.4)

– 3 –

JCAP07(2013)049

the event rate is given by

R(Enr, t) = C F 2(Enr) η(vm, t) with C =ρχσ

0A

2mχµ2χA

. (2.5)

The coefficient C contains the particle physics dependence, while η(vm, t) parametrizes theastrophysics dependence. The halo integral η(vm, t) is the basis for the astrophysics indepen-dent comparison of experiments [20, 21] and we will make extensive use of it below.

3 Bound on the annual modulation amplitude from the expansion in ve

In ref. [28] some of us have derived an upper bound on the annual modulation amplitude interms of the unmodulated rate. Here we briefly review the idea and generalize the bound tothe case of inelastic scattering, where special care has to be taken about the validity of theexpansion.

The DM velocity distribution in the rest frame of the Sun, f(v), is related to the distri-bution in the detector rest frame by fdet(v, t) = f(v + ve(t)). The basic assumption of [28]is that f(v) is constant in time on the scale of 1 year and is constant in space on the scaleof the size of the Sun-Earth distance. These are very weak requirements, called “Assump-tion 1” in [28], which are expected to hold for a wide range of possible DM halos. Thoseassumptions would be violated if a few DM substructures of ∼ 1 AU in size would domi-nate the local DM distribution. The smallest DM substructures in typical WIMP scenarioscan have masses many orders of magnitude smaller than M�, e.g. [29]. Based on numericalsimulations it is estimated in [30] that Earth mass DM substructures with sizes comparableto the solar system are stable against gravitational disruption, and on average one of themwill pass through the solar system every few thousand years, where such an encounter wouldlast about 50 years. Those considerations suggest that Assumption 1 is well satisfied. Letus stress that typical DM streams or debris flows [31] which may dominate the DM halo athigh velocities (especially relevant for inelastic scattering) are expected to be many ordersof magnitude larger than 1 AU, and the relevant time scales are much larger than 1 yr, andhence they fulfill our assumptions, see e.g. [32] and references therein.

Under this assumption the only time dependence is due to the Earth’s velocity ve(t),which we write as [33]

ve(t) = ve[e1 sinλ(t)− e2 cosλ(t)] , (3.1)

with ve = 29.8 km/s, and λ(t) = 2π(t − 0.218) with t in units of 1 year and t = 0 atJanuary 1st, while e1 = (−0.0670, 0.4927,−0.8676) and e2 = (−0.9931,−0.1170, 0.01032) areorthogonal unit vectors spanning the plane of the Earth’s orbit which at this order can beassumed to be circular. The DM velocity distribution in the galactic frame is connected to theone in the rest frame of the Sun by f(v) = fgal(v+vsun), with vsun ≈ (0, 220, 0) km/s +vpec

and vpec ≈ (10, 13, 7) km/s the peculiar velocity of the Sun. We are using galactic coordinateswhere x points towards the galactic center, y in the direction of the galactic rotation, and ztowards the galactic north, perpendicular to the disc. As shown in [34], eq. (3.1) provides anexcellent approximation to describe the annual modulation signal.

Using the fact that ve is small compared to other relevant velocities, one can expandthe halo integral eq. (2.4) in powers of ve. At zeroth order one obtains

η0(vm) =

∫v>vm

d3vf(v)

v, (3.2)

– 4 –

JCAP07(2013)049

which is responsible for the unmodulated (time averaged) rate up to terms of order v2e . The

first order terms in ve lead to the annual modulation signal, which due to eq. (3.1) will havea pure sinusoidal shape, such that

η(vm, t) = η0(vm) +Aη(vm) cos 2π[t− t0(vm)] +O(v2e) , (3.3)

where the amplitude of the annual modulation, Aη(vm), is of first order in ve.In [28] it has been shown that under the above stated “Assumption 1” the modulation

amplitude is bounded as

Aη(vm) < ve

[− dη0

dvm+η0

vm−∫vm

dvη0

v2

]. (3.4)

From eq. (3.2) it is clear that η0 is a positive decreasing function, i.e., dη0/dvm < 0. Asmentioned above, in the case of inelastic scattering typically only a small range in minimalvelocities vm is probed. We denote this interval by [umin, umax] with ∆v = umax − umin. Theboundaries of this interval are determined by the threshold of the detector on one side andby the galactic escape velocity or the nuclear form factor suppression on the other side. Forinelastic scattering ∆v is small. It will thus be convenient to integrate the inequality (3.4)over the interval [umin, umax]. By changing the order of integrations of the double integralwe find∫ umax

umin

dvAη(v) < ve

[η0(umin)− η0(umax) + umin

∫ umax

umin

dvη0

v2−∆v

∫umax

dvη0

v2

]< ve

[η0(umin) + umin

∫ umax

umin

dvη0

v2

]. (3.5)

Integrating again over umin we obtain∫ umax

umin

dvAη(v)(v − umin) <ve2

∫ umax

umin

dvη0

(3− u2

min

v2

)(3.6)

<ve2

(3− u2

min

u2max

)∫ umax

umin

dvη0 . (3.7)

Hence we obtained a bound on the integral of the annual modulation in terms of an integralof the unmodulated rate at first order in ve.

2 This bound receives no corrections at orderO(v2

e) and hence is valid up to (but not including) terms of order O(v3e) [35]. In applying

eq. (3.7) we use only an upper bound, ηbnd on the unmodulated signal η0, allowing alsofor the presence of background. However, we assume that Aη is background free, i.e., thebackground shows no annual modulation.

Let us define the average over the velocity interval by

〈X〉 =1

∆v

∫ umax

umin

dvX(v) . (3.8)

Estimating∫ umax

umindvAη(v)(v− umin) ∼ ∆v

∫ umax

umindvAη(v) and neglecting O(1) coefficients we

obtain from eq. (3.7)

〈Aη〉 .ve∆v〈η0〉 . (3.9)

2Below we will use the bound (3.7) for the numerical analysis since this will allow for easy comparisonwith the “trivial bound” discussed later. The numerical difference between the bounds using (3.7) or (3.6) issmall, typically less than 10%.

– 5 –

JCAP07(2013)049

Figure 2. Contours of the ratio ve/∆v as a function of DM massmχ and mass splitting δ for scatteringon iodine in DAMA, assuming that ∆v is the overlap between the velocity ranges corresponding tothe DAMA [2, 4] keVee energy range for scattering on iodine and the XENON100 [6.61, 43.04] keVrange. Dashed curves show contours of constant vm = 400, 600, 800 km/s, with vm being the minimalvelocity corresponding to the above energy interval for the given δ and mχ.

This shows that the expansion parameter in deriving the bound (3.7) is ve/∆v. In contrastto expressions like ve/umin which are always small, the ratio ve/∆v can become of order one,in particular for inelastic scattering.

As an example we show in figure 2 the ratio ve/∆v as a function of the DM mass mχ andthe inelasticity parameter δ. Elastic scattering is recovered for δ = 0. The velocity interval∆v is chosen having in mind a possible explanation for DAMA. The signal in DAMA ispredominantly in the energy region [2, 4] keVee. As explained in section 4 below, we take for∆v the overlap between the velocity ranges corresponding to the DAMA [2, 4] keVee intervalfor scattering on iodine and the XENON100 [6.61, 43.04] keV interval. From figure 2 one seesthat the expansion parameter ve/∆v & 1 for a non-negligible part of the parameter spaceof DM masses mχ and mass splittings δ. For those values of mχ and δ the bound eq. (3.7)does not apply. Still, in a significant part of the parameter space ve/∆v is sufficiently smallsuch that the expansion can be performed. In particular, we observe from the figure thatfor elastic scattering (δ = 0) and mχ . 50 the expansion parameter is small, justifying theapproach of ref. [25].

4 Halo-independent tests for inelastic scattering

In this section we present three different halo-independent tests of the tension between theDAMA annual modulation signal [2] and the bound from XENON100 [8] in the frameworkof inelastic scattering. The tests are presented in the following order. First, we present theshape test which is a test based on the predicted shape of the signal. Second, we present thebound on the annual modulation signal from eq. (3.7). Third, we present the trivial bound

– 6 –

JCAP07(2013)049

which is based on the fact that the amplitude of the annual modulation must be smaller thanthe unmodulated rate.

The vm method [20, 21] to compare different experiments like DAMA and XENON100requires to translate the physical observations in nuclear recoil energy Enr into vm spaceusing eq. (2.2). Then experiments can be directly compared based on the halo integralη(vm) or inequalities such as eq. (3.7) [25]. However, for inelastic scattering this involvessome complications. The reason is that in inelastic scattering each minimal velocity vm cancorrespond to up to two values of Enr, depending on the values of mχ and δ. This has to betaken into account when translating an observation at a given Enr into vm, since the relationbetween them is no longer unique (as it is for elastic scattering). Solving eq. (2.2) for Enr,one obtains two solutions E± as a function of vm,

E± =

(µχAmA

)[(µχAv

2m − δ)± vm

√µχA(µχAv2

m − 2δ)]. (4.1)

There is a minimal value of vm given by√

2δ/µχA at an energy Emin = µχAδ/MA.

Let us consider the following situation, having in mind DAMA: we have a region [E1, E2]in nuclear recoil energy where the modulation amplitude is non-zero. We assume that E1 isthe threshold energy of the detector. When mapped into vm space according to eq. (2.2) weobtain that the whole interval [E1, E2] is mapped into a small region in vm, between umin

and umax with umax − umin � umin, where umin and umax are the minimum and maximumvalues of vm in the [E1, E2] interval, respectively. For the special case plotted in figure 1,umin = vm(Emin), and umax = vm(E1). In general, depending on the shape of vm as a functionof Enr in the interval [E1, E2], umax may either be vm(E1) (as in the case shown in figure 1)or vm(E2). Furthermore, in cases where Emin falls outside of the interval [E1, E2], umin willeither be vm(E1) or vm(E2). We will not discuss all these cases here explicitly, but as anexample focus on the case shown in figure 1.

To compute the bound in eq. (3.7) using DAMA data, we need to numerically computeintegrals such as

∫ umax

umindvh(v)Aobs

η (v) where h(v) = v−umin is specified in eq. (3.7) (we leaveit general here to apply the same formalism also to the bound in eq. (4.7) discussed later on,where h(v) = 1) and Aobs

η (v) is the observed amplitude of the annual modulation in units ofevents/kg/day/keV. In order to compute those integrals we have to consider the functionalrelation between vm and Enr in the relevant interval [E1, E2]. Let us discuss for instance thesituation depicted in figure 1. In this case we have∫ umax

umin

dvh(v)Aobsη (v) =

∫ umed

umin

dvh(v)Aobsη (v) +

∫ umax

umed

dvh(v)Aobsη (v)

=

∫ umed

umin

dvh(v)Aobsη (v) +

∫ E1

Emed

dEnrdv

dEnrh(Enr)A

obsη (v) . (4.2)

Here, umed = vm(E2) and Emed = E−(umed). The integrals can be written as a sum of severalintegrals which are evaluated over energy bins, as given by the DAMA binning. We take fourbins of equal size in the [2, 4] keVee range for the DAMA data. In each bin we write [25]

Aobsη (vi) =

Aobsi qI

A2IF

2I (Enr)fI

, (4.3)

– 7 –

JCAP07(2013)049

Figure 3. The DM exclusion regions (red bands, with significance as denoted) that follow from theinternal consistency shape test for DAMA data, see eq. (4.4). In the gray region denoted by “no shapetest” there is a one-to-one correspondence between Enr and vm since Emin lies outside the relevantenergy interval [E1, E2] and therefore the shape test cannot be applied.

where the index i labels energy bins, qI is the iodine quenching factor for which we takeqI = 0.09,3 FI(Enr) is the Helm form factor for iodine, and fI = mI/(mNa + mI). Ineach energy bin we assume Aobs

i is constant, and thus in each bin we numerically integrate(dv/dEnr)h(Enr)/F

2I (Enr) over the bin width.

For the first integral on the r.h.s. of eq. (4.2) there is an ambiguity, since the interval[umin, umed] corresponds to two regions in energy: [Emed, Emin] or [Emin, E2]. If the inelasticDM hypothesis under consideration is correct, both energy intervals should give the samevalue of the integral. We can use this observation to test the hypothesis that the signal isdue to inelastic DM scattering by requiring that the two integrals agree within experimentalerrors. In the following, we will call this the “shape test”. Let us denote the integralscorresponding to the two energy intervals by Ia and Ib and their experimental errors by σaand σb, correspondingly. In figure 3 we show the difference weighted by the error as obtainedfrom DAMA data:

|Ia − Ib|√σ2a + σ2

b

. (4.4)

We observe that a strip in the parameter space in δ and mχ is already excluded by thisrequirement at more than 3σ in a completely halo-independent way, just requiring a spectralshape of the signal consistent with the inelastic scattering hypothesis. In cases where thetwo values are consistent within errors we use for the integral the weighted average of thetwo values. In figure 3, Ia and Ib are evaluated for the choice of h(v) = v − umin. The shape

3For DM masses that we consider one can safely neglect scattering on sodium. Note also that the channelingfraction of iodine in NaI is likely to be very small and can be neglected [36].

– 8 –

JCAP07(2013)049

test is only slightly different for h(v) = 1 which is the case for the trivial bound explainedlater in eq. (4.7).

In order to evaluate the r.h.s. of the inequality in eq. (3.7), we need to calculate anintegral over the experimental upper bound ηbnd(vm) on the unmodulated signal, with

η(vm) ≡ σpρχ2mχµ2

χp

η0(vm) , (4.5)

where σp is the cross section on a nucleon and µχp is the DM-nucleon reduced mass. ηhas units of events/kg/day/keV. In using eqs. (4.3) and (4.5) we have assumed an A2

dependence of the scattering cross section on the nucleus with mass number A. We usethe method discussed in ref. [25] (see also [20]) to evaluate ηbnd(vm) for the inelastic case.Namely, we use the fact that η(vm) is a falling function, and that the minimal number ofevents is obtained for η constant and equal to η(vm) up to vm and zero for larger values ofvm. Therefore, for a given vm we have a lower bound on the predicted number of events inan interval of observed energies [E1, E2], Npred

[E1,E2] > µ(vm) with

µ(vm) = MTA2η(vm)

∫ E+

E−

dEnrF2A(Enr)G[E1,E2](Enr) , (4.6)

where G[E1,E2](Enr) is the detector response function which describes the contribution ofevents with the nuclear-recoil energy Enr to the observed energy interval [E1, E2]. M andT are the detector mass and exposure time, respectively. Notice that µ(vm) for the elasticcase is given in eq. (10) of ref. [25] and in that case the integral is computed between 0 andE(vm) which corresponds to velocities below a fixed vm. For the inelastic case, we have twosolutions E+ and E− for each vm, and the region in velocity space below vm is precisely, theregion E− < Enr < E+.

Assuming an experiment observes Nobs[E1,E2] events in the interval [E1, E2], we can obtain

an upper bound on η(vm) for a fixed vm at a confidence level CL by requiring that theprobability of obtaining Nobs

[E1,E2] events or less for a Poisson mean of µ(vm) is equal to 1−CL.

The upper bound obtained in this way is ηbnd(vm) and can then be used in eq. (3.7) andnumerically integrated over [umin, umax] to constrain the modulation amplitude. We use thedata from XENON100 [8] where the Enr interval [6.61, 43.04] keV is binned into four bins.In each bin we calculate the probability of obtaining Nobs

[E1,E2] events or less for a Poisson

mean of µ(vm) as described above, and then multiply the probability of the four bins toobtain the overall probability, giving finally the actually observed event distribution. Note,that since only the high energy range in Xenon is relevant, our results are not sensitive touncertainties in the scintillation efficiency Leff at low energies. For the comparison of theDAMA and XENON100 data using eq. (3.7) we define the [umin, umax] range as the overlapbetween vm spaces corresponding to the DAMA iodine [2, 4] keVee range and the XENON100[6.61, 43.04] keV range.4

In figure 4 we show the l.h.s. and r.h.s. of the bound from eq. (3.7) in red and black,respectively, as a function of mχ for δ = 50 keV, 100 keV, 120 keV, and 150 keV. We calculatethe integral over the annual modulation amplitude in the l.h.s. of eq. (3.7) as described above.The red dashed curves indicate the 1σ error on the integral. The upper limit on the r.h.s.of eq. (3.7) is calculated from the XENON100 3σ upper limit. We see that in most regions

4For most of the region in parameter space this joint interval is actually very close to the one coming fromDAMA iodine [2, 4] keVee.

– 9 –

JCAP07(2013)049

l.h.s. HtrivialL

r.h.s. Hupper boundL

l.h.s.

20 50 100 200 500 100010-6

10-5

10-4

10-3

10-2

mΧ HGeVL

Iodine, ∆=50 keV, General bound


l.h.s. HtrivialL

l.h.s.

20 50 100 200 500 100010-5

10-4

10-3

10-2

mΧ HGeVL


l.h.s. HtrivialL


l.h.s.

20 50 100 200 500 100010-5

10-4

10-3

10-2

mΧ HGeVL


l.h.s. HtrivialL


l.h.s.

20 50 100 200 500 100010-5

10-4

10-3

10-2

mΧ HGeVL


Figure 4. The bounds from eqs. (3.7) and (4.7) for DAMA data as a function of mχ for δ =50, 100, 120, 150 keV. The red curve labeled “l.h.s.” shows the integral on the l.h.s. of eq. (3.7), whereasthe blue curve labeled “l.h.s. (trivial)” corresponds to the l.h.s. of the trivial bound in eq. (4.7). Thedashed curves indicate the 1σ error. The black curve labeled “r.h.s. (upper bound)” is the samefor (3.7) and (4.7) and has been obtained from the 3σ limit on ηbnd from XENON100 data. The unitson the vertical axis are counts/kg/day/keV (km/s)2. In the gray shaded regions we have ve/∆v > 0.7;truncating the expansion may not be a good approximation and hence, the red curve should not betrusted in those regions, but instead the blue one can be used there. The solid (dashed) vertical linesindicate the regions where the two integrals relevant for the “shape test” differ by more than 2σ (3σ)according to eq. (4.4).

of the parameter space the bound is strongly violated, disfavoring an inelastic scatteringinterpretation of the DAMA signal halo-independently. Note that DM mass enters only viaµχA, so that µχA ' mA for mχ � mA, and eq. (4.1) becomes independent of mχ. This iswhat we see in figure 4, where curves become flat for large mχ and therefore the tensionbetween XENON100 and DAMA cannot be diminished when going to larger DM masses.

The shaded regions in figure 4 are the regions where the expansion parameter ve/∆vis large (we take somewhat arbitrarily ve/∆v > 0.7, cf. also figure 2). Hence, in the shadedregions the astrophysics independent bound on the modulation amplitude, eq. (3.7) (the redcurves in figure 4), can receive O(1) corrections and should not be trusted. Interestingly,however, part of the region where the ve expansion breaks down is disfavored by the internalconsistency “shape test”, eq. (4.4). We indicate the range in mχ where the two integralsdiffer by more than 2σ and 3σ with vertical lines, cf. also figure 3.

– 10 –

JCAP07(2013)049

Finally we consider the “trivial bound”, which is based on the simple fact valid for anypositive function that the amplitude of the first harmonic has to be smaller than the constantpart, i.e., Aη ≤ η0. To compare directly with eq. (3.7), we can write the trivial bound as,

ve2

(3− u2

min

u2max

)∫ umax

umin

dvAη(v) <ve2

(3− u2

min

u2max

)∫ umax

umin

dv ηbnd(v) . (4.7)

The l.h.s. of this relation is shown as blue curve in figure 4 together with its 1σ error band.We observe again strong tension with the upper bound from XENON100 data. Clearly thisbound is independent of any expansion parameter and is valid in the full parameter space.In the regions where the expansion is expected to break down (i.e., the grey shaded regions)this bound can be used to exclude the inelastic explanation for DAMA. From figure 4 we alsoobserve that in the regions where the expansion in ve is expected to be valid the modulationbound from eq. (3.7) becomes stronger (or at least comparable — for large mχ) to the trivialbound.

5 Conclusions

Inelastic scattering [9] has originally been invoked to reconcile the DAMA annual modulationsignal with bounds from other experiments. Using the kinematics of inelastic scattering theannual modulation amplitude can be enhanced compared to the time averaged rate. This isachieved at the expense of tuning the minimal velocity probed by the experiment so that itis close to the galactic escape velocity, which makes the signal rather sensitive to propertiesof the tails of the dark matter velocity distribution. Hence it is important to establishhalo-independent methods for this scenario.

In this work we have generalized the comparison of dark matter direct detection ex-periments in vm space [20, 21] to the case of inelastic scattering. This is non-trivial due tothe non-unique relation between the recoil energy and vm. Turning this complication intoa virtue, we presented a consistency check based on the particular shape of the signal forinelastic scattering which we dubbed the “shape test”, given in eq. (4.4) and figure 3. Incertain regions of the parameter space the inelastic scattering hypothesis can be excludedsimply based on the energy spectrum of the modulation signal, without referring to haloproperties.

Furthermore, we have applied a bound on the annual modulation amplitude based on anexpansion of the halo integral in the Earth’s velocity ve [28]. We have identified the relevantexpansion parameter to be ve/∆v, where ∆v is the range in minimal velocities vm probed inthe experiment. For inelastic scattering, ∆v can become of order ve for part of the (mχ, δ)parameter space, and then the bound cannot be applied. However, in those cases one canuse the “trivial bound”, requiring that the amplitude of the annual modulation has to beless than the bound on the unmodulated rate.

We were able to show that XENON100 strongly disfavors an interpretation of the DAMAmodulation signal in terms of inelastic scattering, independent of assumptions on the prop-erties of the local dark matter velocity distribution. Beyond the immediate problem of inter-preting the DAMA signal, the methods developed in this manuscript will provide a valuableconsistency check for an inelastic scattering interpretation of any future dark matter signal.

In our work we have focused on spin-independent contact interactions, where the dif-ferential scattering cross section takes the form of eq. (2.3). Our considerations generalize

– 11 –

JCAP07(2013)049

trivially to other interaction types which lead to a similar 1/v2 dependence (e.g., the spin-dependent inelastic scattering considered in [15]) but may feature a different dependenceon Enr. Furthermore, the shape test and the trivial bound can be applied for any particlephysics model where the differential cross section factorizes as X(v)Y (Enr), where X and Yare arbitrary functions of v and Enr, respectively. An example where such a factorization isnot possible in general are magnetic interactions, e.g. [37]. In such cases generalized methodsas presented recently in [38] may be invoked.

Acknowledgments

N.B., J.H.-G., and T.S. acknowledge support from the European Union FP7 ITN INVISI-BLES (Marie Curie Actions, PITN-GA-2011-289442). J.H.-G. is supported by the MICINNunder the FPU program. J.Z. was supported in part by the U.S. National Science Foundationunder CAREER Grant PHY-1151392.

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[31] M. Kuhlen, M. Lisanti and D.N. Spergel, Direct detection of dark matter debris flows,Phys. Rev. D 86 (2012) 063505 [arXiv:1202.0007] [INSPIRE].

[32] K. Freese, M. Lisanti and C. Savage, Annual modulation of dark matter: a review,arXiv:1209.3339 [INSPIRE].

[33] G. Gelmini and P. Gondolo, WIMP annual modulation with opposite phase in Late-Infall halomodels, Phys. Rev. D 64 (2001) 023504 [hep-ph/0012315] [INSPIRE].

[34] A.M. Green, Effect of realistic astrophysical inputs on the phase and shape of the WIMPannual modulation signal, Phys. Rev. D 68 (2003) 023004 [Erratum ibid. D 69 (2004) 109902][astro-ph/0304446] [INSPIRE].

– 13 –

JCAP07(2013)049

[35] N. Bozorgnia, J. Herrero-Garcia, T. Schwetz and J. Zupan, in preparation.

[36] N. Bozorgnia, G.B. Gelmini and P. Gondolo, Channeling in direct dark matter detection I:channeling fraction in NaI (Tl) crystals, JCAP 11 (2010) 019 [arXiv:1006.3110] [INSPIRE].

[37] S. Chang, N. Weiner and I. Yavin, Magnetic inelastic dark matter,Phys. Rev. D 82 (2010) 125011 [arXiv:1007.4200] [INSPIRE].

[38] E. Del Nobile, G. Gelmini, P. Gondolo and J.-H. Huh, Generalized halo independentcomparison of direct dark matter detection data, arXiv:1306.5273 [INSPIRE].

– 14 –

Philosophical reflection

Since I have use of reason, I have always been asking myself questions aboutthe world and wanted to learn everything about Nature. Some of them wentso deep inside my mind that really obsessed me and sometimes made me feellost. The thoughts, extended from childhood to nowadays, which I could callthe dialogue of my life, are something like this:

∙ What is all this?

∙ You mean the world, etc...?

∙ Yes... all this complexity we see around us.... the world... I know it is aplanet inside a spiral galaxy, just like many others of the billions thatare out there, in a Universe that is flat, expanding, in fact accelerating...but apart from this, we don’t know much...

∙ We know a lot, what do you mean?

∙ Come on, you did not even read the introduction...!! We know about5% of the energy content of the Universe! We are here in a Universewhose more abundant ingredients, like dark energy or dark matter wehave not detected, we have no idea what makes most of it...

∙ OK, one step at a time... what bothers you?

∙ Why all this... what are we... I mean, I know we are a human beings,not so different from other animals: made out of carbon through millionsof years of evolution, conscious about our existence, and more intelligentthan other mammals, with some abstract thinking... We have justevolved from inert matter, probably from some primary amino-acidsthat were made thanks to lightning... We are alive in the sense that weare born, we build ourselves with inert matter, atoms that are most ofthem made up in the stars, we reproduce so the species survives, we dieand we become inert again... but.... where do we come from?

ii Philosophical reflection

∙ That one I know, from the Big Bang ∼15.000.000.000 years ago, andfrom there on... before no time can be defined... In fact, we have adecent understanding of what happened from ∼ 10−10 seconds after ituntil now, which looks quite amazing... although it is true that in energyscale, we are several orders of magnitude below the Planck mass....

∙ but I think it is an allowed to question to ask why.... why all this?

∙ Why? Come on...!! You just ask how!!! Otherwise you should go to thephilosophy school then and do some bla bla bla...

∙ You mean a physicist should not look for the reason why things happen,be happy with just a phenomenological description of how things work...?I believe the separation between why and how is not so well defined...and I want to use the scientific method, test my theories and rejectthem if proven to be false!!

∙ OK... You know, we are just the result of evolution... let’s say4.000.000.000 years ago life started, random mutations and evolutionget the fittest to survive, reproduce and therefore to evolve and we arehere, with our brothers the chimpanzees... so we have just the brainand senses needed to understand how some things work to be able tosurvive, live a couple of years, but not much more... In fact some peopleseem to have just the brain to pass the day :)...

∙ I guess that is the reason why when deadlines approach my brain worksbetter, because then it is no longer a matter of curiosity but of survival...:) but really, why does everything exist?

∙ The why questions seem beyond our capacities... I guess we ask wheredo all atoms come from? from stars... and why do stars shine? andwe can find an answer... because of nuclear fusion... but why arethere protons? Because of QCD’s phase transition... and Heliumnuclei? Nucleosynthesis in the Early Universe, 3 minutes after the BigBang... and when do Hydrogen atoms form? at recombination, belowHydrogen’s binding energy... in fact the CMB is a snapshot of 380.000years after the Big Bang, when protons and electrons combined to formhydrogen, that is a pretty amazing picture of the Big Bang... and quarksand leptons? Probably some GUT, SO(10)... and three families? someflavour symmetry... but why these symmetries? String Theory... and soon... but why does the Universe exist?

Philosophical reflection iii

∙ I deeply think there is a logic in all this... Quantum fluctuations ofsome field... we have even recently discovered gravitational waves frominflation! We are speaking about times as early as ∼ 10−36 seconds... inany case, it is true that without knowing what are dark matter and darkenergy, and the inflationary model, and a quantum theory of gravity,we seem to be far from this question....

∙ so you really think that everything can be described in a mathematicalway...

∙ Mathematics are just a language, a tool... I don’t believe in a deepmeaning of a perfect Platonic triangle... we invent mathematics, somedo describe Nature extremely well, some are good approximations of itand some have nothing to do with it... but there always seem to be awhy question to which we have no clue...

∙ So we give up and enjoy life...

∙ What!!??

∙ We said it was going to be hard, we have begun the soccer match withthe leg injured, just one eye, no contact lenses and with a headache,but for sure there is a logic in all this mess, a pattern, a symmetry thatgot broken by quantum fluctuations in the Big Bang and gave all thisapparent non-sense... (don’t ask me why that symmetry please...) butdon’t be pessimistic, we know a lot, even if it is true that we don’t knowmost of it... from a working perspective it is good news: there are manythings to investigate...

∙ Yes, you are right... So what do we do?

∙ We continue... being smarter... working harder.... we should buildan enormous accelerator, precise satellites, super-pure undergroundexperiments... As you know, one can do physics in several differentways:

– Build experiments without a particular motivation, trying to see ifNature reveals any of its mysteries.

– Based on some theoretical and/or experimental motivation, proposea complete new theoretical framework, with a strong mathematicalbackground needed, and try to test it, probably in the very longrun, as you are in the frontier of theory/experiment.

iv Philosophical reflection

– Try to explain some experimental result which is not understood,proposing a new model with new particles, interactions... whichhopefully will give signals in other experiments, so it can be testedin the near future.

– and so on...

Choose your preferred way, and go for it...

∙ OK, I like the idea of proposing testable models, and trying to under-stand strange experimental results...

∙ So in which topic would you like to do your research? The time inHistory is important... maybe we don’t have the knowledge, or it is notthe time, to solve the flavour puzzle... or to unify gravity and quantumphysics, remember that not even Einstein could, and that guy was kindof smart... and these theories can be difficult to test... Neutrinos haverecently been discovered to have masses and we still don’t know howthey get them...

∙ That sounds really interesting...!

∙ But keep and eye on dark matter, it could be the next one to be discov-ered. Its experimental situation looks pretty much like the confusingsituation with neutrino oscillations some years ago...

∙ That sounds also great!

∙ Good luck!

Agradecimientos

Nunca habría podido realizar esta tesis sin la educación, guía y ayuda quemuchas personas me han brindado. Algunos no han participado directamente,pero mi salud mental y estado de ánimo depende de ellos, y repercutedirectamente en la ciencia que hago, así que todo el mundo que aquí aparece,y otra mucha gente que me habré dejado, ¡me ha ayudado de una manera uotra!

Quiero empezar por mis padres, para los que mi educación ha sido yes lo más importante. En concreto, desde su experiencia, me han dadobuenos consejos, como que tan importante o más que que rodearte de gentecompetente, es hacerlo de buenas personas. Creo que de mi madre he sacadocomo virtudes científicas la pragmaticidad, la eficiencia, la enorme capacidadde trabajo cuando estamos motivados, el ser echado para adelante y micapacidad de socialización, y de mi padre la capacidad de abstracción, laimaginación, la profundización a largo plazo en un tema, la inteligencia einterés a la hora de aprender algo nuevo, y el intentar siempre ver el bosquesin quedarse únicamente en los árboles, buscando la perspectiva y la visión deconjunto. La mezcla ideal no es fácil de conseguir, todo tiene pros y contras,y a veces se peca de correr, y otras de divagar... ¡pero estoy contento de lalotería genética!

A mis tías quiero agradecerles que, aunque nunca he conseguido queentiendan lo que hago, siempre me han apoyado, y ponían velas en losexámenes. También quiero agradecer a mis tíos y primos, por apoyarme,interesarse y animarme, y porque siempre me encuentro muy a gusto conellos. En especial a mi tío Alejandro, por interesarse mucho por la física, ypor decirme en el bachiller (más allá de que se compinchara con mis padres yotros tíos míos para que hiciera medicina) que física no era una carrera, erala carrera. Y también quiero recordar a mi tío Ramón, con el que conversabade ciencia en las comidas familiares, y al que se le echa de menos.

En segundo lugar quiero agradecer a mis directores de tesis, Arcadi yNuria, que creo que me han guiado con acierto en esta andadura. Con ellos eltrato ha sido siempre más que bueno y fácil, y las puertas de sus despachos

vi Agradecimientos

siempre han estado abiertas. De ambos he aprendido muchas cosas. Ademásquiero agradecerles la lectura de esta tesis y las sugerencias que me han hecho.

De Nuria espero haber aprendido su seriedad y rigor científicos, su espíritucrítico, por ejemplo analizando en profundidad un artículo, y su detallismo alescribir un paper, pensando con cuidado cada frase, y le agradezco su buenadirección. Además siempre se ha preocupado por las mil cosas que conllevahacer una tesis (y que parece que no están, pero que están), desde el millónde burocracias torturadoras a las decisiones de estancias, de dar charlas, deconferencias, visados, informes, memorias.. y un largo etc.

De Arcadi me conformo con que se me haya pegado algo de su capacidadmatemática e intuición física, y de su tesón y resistencia al abandono, elpicarse hasta que se resuelve la maldita integral o lo que sea. Le quieroagradecer las charlas de física en la pizarra de su despacho, y las miles dedudas que siempre me ha resuelto, incluyendo entre ellas una buena dosis deinformáticas, así como su preocupación en las distintas facetas asociadas a larealización de una tesis.

En fin, ¡muchas gracias a los dos!Y mención especial también a Thomas, el cual me acogió en Heidelberg

con los brazos abiertos, y me introdujo en un muy buen tema de candenteactualidad. Con él he tenido la oportunidad de seguir colaborando y esperoseguir haciéndolo en mi nueva andadura. Ojalá se haya producido ciertaósmosis hacia mi persona de su increíble capacidad en todo lo relacionadocon el trabajo científico. Para mí ha sido casi como un tercer supervisor, yno puedo más que agradecérselo.

También quiero agradecer a André la posibilidad de haber ido a Chicago,estancia que me resultó muy provechosa en todos los aspectos. Respecto a lafísica me permitió introducirme en el mundo de las teorías efectivas, que mehan resultado muy interesantes. Además quiero agradecerle la ayuda que mebrindó para dar charlas, por ejemplo en Fermilab, donde disfruté mucho delgrupo tan activo que hay en física.

A todos ellos quiero además agradecerles sus cartas de recomendación ala hora de solicitar postdoc, que he conseguido.

Quiero también agradecer a los miembros del tribunal la lectura de estatesis y las sugerencias que me han hecho. En especial a Sergio y Laura, queme han dicho una buena cantidad de typos y de útiles sugerencias.

Y también hacer una mención especial al resto de personas con las que hepodido colaborar, Alberto, Jure, Nassim, Andrew y Miguel, con los que hetrabajado muy a gusto.

Especialmente agradecido estoy a Alberto, que siempre ha estado allí volun-tarioso, más que disponible para resolverme dudas de todo tipo, especialmentelatexianas, de consola, de mac... etc.

Agradecimientos vii

Acto seguido quiero nombrar a mi grupo: a Pilar, Carlos y Olga, por lasayudas de todo tipo y las conversaciones mantenidas, y porque siempre mehan tratado como uno más. Sobre todo quiero agradecerles todo lo que hedisfrutado atendiendo a las Trobadas, hablando de física con ellos de tú atú... ha sido un placer. Y a Carlos también agradecerle las divertidas rutasgastronómicas en las conferencias.

Del departamento quiero mencionar especialmente a Vicente Vento y aÓscar, con los que he compartido asignatura y ha sido muy fácil y sencillo, ysiempre me han ayudado. A Vicente también le agradezco su carisma, lasrisas y la vida que le da al departamento.

Al resto del departamento y del IFIC, en los que me he sentido como encasa. Ha sido un privilegio y una suerte el haber estado aquí con vosotroshaciendo el máster y el Doctorado.

También quiero destacar la ayuda, consejos, y charlas de física de ungran número de colegas (y amigos) del mundo de la física: Mattias, Enrique,que conocí en Moriond, Laura, Philipp, Yasutaka, Kher Sham... etc, conlos que me lo pasé muy bien en Heidelberg, y una larga lista de colegas yamigos físicos, a bote pronto: Ana Rodrguez, Juan González, Bogna, Joan,Miki y Miguel (Cargese estuvo genial), Rodrigo, Florian, Jacobo, Pilar, JuanRacker, Jordi Salvado idem en Florencia, Martín en Argonne, Ignacio, Elena,Valentina en Durham, Avelino, Javier Redondo, Walter y los Madrid-Bayernexitosos, Tommy, Belén Gavela, Sergio Palomares... etc.

Dicho ésto, quiero remontarme a mis principios en la ciencia. Y creo quehay dos nombres propios que me han marcado en Primaria-Secundaria, y enel Bachillerato, y que me han llevado por el camino correcto. El primero es miprimera profesora de matemáticas y de ciencias (aunque es química) MaríaPilar Alegre1, la cual me hizo entender lo que era el rigor, aprendí mucho y viqué era la ciencia, aunque fuera orientada más a la química y la formulación.

El segundo nombre propio en orden cronológico es Miguel Ángel Per, miprofesor de física en el bachiller, físico teórico, y el que creo que me dió elempujón definitivo para estudiar física: lo explicaba todo tan bien, que conatender en clase, que me encantaba, y con una lectura de los apuntes y unpar de ejercicios ibas al examen y sacabas un 10. Además le hacías preguntas,razonaba, y a veces incluso se paraba a pensar, dudaba... una novedad para

1No quiero dejar de agradecer a Javier su paciencia conmigo, como me decía cuandono callaba en clase, un hijo único, y a Elena su empeño en que escribiera bien: me haríaescribir varias decenas de cuadernillos Rubio. También a María Pilar Gómez, que era purocorazón.

viii Agradecimientos

mí.2 También llegaba a clase, veía una barbaridad3 de otro profesor cuyonombre no diré y cuya preparación y esfuerzo no comentaré, y lo corregía.Allí por fin me di cuenta de que había salida más allá de la memoria: ¡elrazonamiento! ¡Y que encima llevaba menos tiempo y era más gratificante einapelable, universal! Además me apoyó para el premio final de bachillerato,y años después en la Universidad hablamos y me animó a que siguiera en lafísica.

De la facultad de Zaragoza, quiero agradecer especialmente a ManuelAsorey, por darme la oportunidad de ir a Southampton, ayudarme en todo, ypor sus consejos más allá de la física. A José Luis Cortés, con el que disfrutémuchísimo en Mecánica Cuántica, y con el que hablé y me ayudó a orientarmepara hacer una tesis. A Vicente Azcoiti, por hacer de Relatividad Generalalgo sencillo y un disfrute, y por el 10 ;). Al resto de profesores que me hanenseñado cosas, como Fernando Falceto, Antonio Seguí, María Luisa Sarsa,Jesús Atencia (por sus clases de física de primero), Jorge Puimedón, Morellón,Rebolledo (por sus cartas para ir a un curso de verano), Juan Vallés, Forniés(porque electromagnetismo la disfruté mucho), García Vinuesa (por hablarnosde posibilidades de futuro y ser accesible), a Ramón Pla (por corregirmesolicitudes en inglés)...

De Southampton solo tengo buenos recuerdos. Quiero agradecer a NickEvans la asignatura que más he disfrutado en mi vida, Particle Physics, cadahora de clase era un auténtico placer. Recuerdo leerme los apuntes antes de irallí, como quien va al cine, a disfrutar como un enano. También a Steve Kingpor sus didácticas clases, a Alexander Belyaev por ofrecerme la oportunidadde que me quedara, a Beatriz de Carlos, cuyo examen sólo logramos hacerlos tres españoles, y por charlas más allá de la física, y en general a todala Universidad y los profesores, que daban apuntes perfectos antes de queempezara el curso, que te permitían atender y no copiar cual chimpancérápido y mal, e incluso leértelos antes de ir a clase y poder ir con dudaspensadas, y que ponían los exámenes antes del curso, de forma que sabían loque tenían que explicar, ¡y en el examen preguntaban cosas que habían dadoo comentado en clase!

También quiero agradecer a José González el tiempo que disfruté enMadrid gracias a la Beca de Introducción a la Investigación del CSIC, fueuna gran experiencia y resultó muy provechosa.

También a mis compañeros de la carrera, Micha, Noe y Ceci. Al primeropor sus llamadas para comentar dudas del día de antes y por ser uno de mis

2Como una vez que le pregunté cómo se medía la entropía de gases y estuvo un ratopensando...

3¡Que las ondas penetraban o no cuerpos dependiendo de su amplitud!

Agradecimientos ix

mejores amigos, a Noe por ser una buena compañía tanto en la carrera comoen Southampton y por supuesto fuera de la física. Y especialmente a Ceci, milgracias por no se cuántos apuntes que me has pasado, por preparar exámenesjuntos, por comentar dudas etc. Creo que todos fuimos un buen equipo. Y aJacobo por los buenos ratos en Southampton.

Quiero mencionar ahora a mis amigos. A los de Zaragoza, en especial aJuan Artigas, con el que mantuvimos mil conversaciones los primeros tiemposde carrera y nos apoyamos mutuamente. A Víctor, porque llevamos vidasparalelas en todo desde los cuatro años, hemos compartido mil cosas y escomo un hermano para mí. Al resto, por los buenos ratos pasados: a Checo,por las mil risas y por su amistad, a Chemi, que aun cree que hago química,y por las anécdotas, como cuando me quitó mi examen (en mitad del examenal ir a preguntar al profesor) ¡y yo no lo encontraba!, a Flo por hacer que lasteclas de mi mac vayan mal, al rematar de cabeza un cubata al ir a cogeralgo del suelo, cuando les estaba torturando con una charla divulgativa defísica en Salou, a Héctor por interesarse por la física, aunque solo se le hayaquedado por algún motivo la teoría del Big Rip (o Gran Desgarro), y porquegracias a él he aprendido técnicas que no me explican en la carrera, como ladel aguante, la de la huida o la del revoloteo, a Pepe por el tiempo en Madridcuando estuve de estancia y le ocupaba su estante en la nevera con pastapara la semana entera, a Mateo por las risas con sus bolis bic enciclopédicosy los exámenes para los que necesitan dos horas de estudio y acababan nopresentándose, y por seguir preguntándome “pero lo que haces... ¿para quésirve?” y yo no conseguir explicárselo, y a Ramiro por ser un gran jugador depóker y ponérmelo difícil, y por las mcuhas risas, a Guti por nuestras salidasde fiesta, por aconsejarme que me opere el tabique y porque cada vez querecuerdo lo de “estoy en Modo” a las 7 de la mañana estando en Salou me vaa dar algo... a Alejandra por nuestras conversaciones... al resto del grupo...a Chicho, Laura, Iñaki, María etc etc... gracias.... A David y Alex, por lascharlas técnicas sobre el póker.... y a todos los demás.

A los de Valencia. A Alberto, por ser un gran amigo, por todo, por lasmil conversaciones que hemos tenido, por haber estado ahí.... A Clarilla,por ser una de las mejores personas que conozco y una gran amiga. A Javi,por ser una gran persona, por las risas, por las fiestas, por los ratos deteoría y análisis de probabilidad del póker en la pizarra de tu despacho...A Oli, tras saber la teoría, porque gracias a él aprendí a jugar al pókerde verdad, con depuradas técnicas como la de la grúa o la del all-in sinver cartas, y por los mil ratos pasados de fiesta y nuestras anécdotas... AAntonio, por los ratos disfrutados juntos tanto en el IFIC como fuera, y porlas conversaciones con unas cervezas, y también por ayudarme con algunaque otra duda informática... A Gammerman, por tu movimentación hasta la

x Agradecimientos

conquista y por las anécdotas.... a Vido, por nuestras conversaciones y porser entrañable, y porque además hiciste que recuperara la fe, a Montse, porla ilusión y por todo... y por tu amistad, que deseo continúe en el futuro...A Mira, por ser divertida y genial... ¡aunque a veces no nos deje dormir! ;)A Joaquín, por el rescate tras el naufragio, y por las historias disfrutadasjuntos... a Beni, por las conversaciones profundas que disfruto, y por lasrisas y los buenos momentos, y por abrirme los ojos al póker de verdad aljugar contra ti y al hablarme de un libro, y a Diana, por estar siempre debuen humor, con una sonrisa en la cara... a Carmen y al resto por los buenosmomentos que hemos pasado... A David Castelló, por ser mi primer amigo enValencia, por las pizzas de los jueves de mi primer año, porque recién llegadome introdujiste en tu círculo valenciano... a Ivana, por habérnoslo pasadomuy bien y por tus detalles... a Andrea, por las risas, y por recordarme quelas resistencias en serie y en paralelo son muy diferentes :) ... a Agustín,por nuestras conversaciones y su buena compañía en las comidas del IFIC, aFrancesca, aunque tenga poca fé en mis posibilidades ;), a María, aunque tehayas ido, por ser una gran amiga con la que lloro de risa... a Carl, por lascenas en el Kin-Ma y nuestras conversaciones... a Ebe, por nuestras largasconversaciones de café en el IFIC, y por ser una confidente espectacular...a Miguel, más allá de la física, por tu sentido del humor particular queameniza las comidas, y por nuestras conversaciones políticas... a Joan (yOli, claro), por los ratos de póker cash en casino y por enseñarme a floatear,a Rubén, Nuria, Aaron (con el que he compartido algunas conferencias, ypor leerse el capítulo de cosmología y sugerirme algunas pequeños cambios),Alex Celis, Alberto Filipuzzi, Elena, Manu, Francesc, Juan Racker, Marija,Marisa, Marga, Luis, Adrián, Rafa, Raquel, Japepe, Joel ... etc por hacermás divertido el trabajo y por lo que hemos compartido juntos.

A los amigos del colegio (Diego, Álvaro, Pedro, Miguel, Laura, Ana... )y más recientemente al equipo de fútbol, al de frontón, al de los torneos depóker de los viernes... por darle diversión a la rutina...

A mis amigos del pueblo... gracias porque siempre que voy parece que nosvimos ayer, desde las cabañas en la prehistoria, pasando por los partidos defútbol, las rutas en bici, el Ronda, los baños en el río, el local, las múltiplesfiestas y risas en pueblos y en Jaca y muchísimas otras vivencias... En fin,siempre disfruto volviendo a las raíces.

A (algunos de) mis compis de piso a lo largo de éstos años, como Tanit,por su tenacidad con los problemas e inteligencia, y porque siempre estásde buen humor, a Lola, por los desfases y nuestras charlas en el sofá de laterraza, a David y a Aritz con los que la convivencia es tan fácil... y a lavecina, sin la cual habría muchas más fiestas en mi casa ¡y por lo tanto menostrabajo!

Agradecimientos xi

A mis amigos de Chicago, a Luiza, por todo y por venirte a Seattleconmigo, a Alejandra, por ser una buena amiga, a Tommasso, porque nosentendemos a la perfección, y por nuestras historias paralelas en Chicago... aGabi, porque eres una grande y me acogiste en tu grupo como uno más, aLuca y a Antonella por los ratos divertidos, a Ben por tu inteligencia social eintuición, sabiendo cosas de mí antes que yo mismo... y a Samara por nuestrasconversaciones... gracias a vosotros mi paso por Chicago fue increíble...

Los que me he dejado y creéis estar, ¡seguro que estáis!Para finalizar, quiero agradecer al difunto Ministerio de Educación, Ciencia

e Innovación (MICINN)4 el permitirme realizar la tesis con una beca FPUy dos estancias de investigación en el extranjero, y en general por apostarpor la investigación y el conocimiento, así como a los distintos proyectos queme han permitido financiar la asistencia a escuelas, congresos y conferen-cias: de la Unión Europea FP7 ITN INVISIBLES, MRTN-CT- 2006-035482(FLAVIAnet); de MINECO: FPA-2007-60323, FPA2011-29678-C02-01 y de laGeneralitat Valenciana: PROMETEO/2009/116.

4Ahora ha descendido de categoría a Secretaría de Estado de Investigación, Desarrollo eInnovación, perteneciente al Ministerio de Economía y Competitividad.

Algunas críticas y sugerencias

Desde mi humildad y experiencia quiero comentar algunos puntos que en miopinión son más que mejorables en el funcionamiento de la educación y elsistema científico en general, y el español en particular.

Primero debo decir que muchos de mis profesores nos obligaban en muchasocasiones a memorizar sin sentido. Hago aquí un llamamiento, al mundoeducativo: no hagan perder el tiempo a los estudiantes memorizando horroresque van a olvidar tras el examen, enséñenles a pensar, argumentar, imaginar...ojalá no hubiera tenido que aprenderme semejantes ladrillos... el que necesitesaber algo de memoria lo estudiará en su momento (por ejemplo en Derechoo Medicina) y siempre puede acudir a las referencias. Practicar y ejercitarla memoria en una dosis prudente puede ser discutible, basar el sistema enrecitar la lección cual loro que no sabe ni lo que dice, no. Lo importante esel razonamiento y el conocimiento, no la erudición. Lo fundamental es lapregunta, no la respuesta.

Y hago un segundo llamamiento de algo que me ha enervado desde eldía uno de mi Doctorado (realmente incluso antes, al pedir becas), que meha frustrado, que mi cabeza no logra comprender: por favor, señores queorganizan las cosas, políticos y burócratas, ¿saben ustedes que nos ahoganen burocracias estúpidas? Es una auténtica barbaridad. No puedo estarhaciendo memorias cada tres meses, para viajes, estancias, al volver, de beca,de contrato... y todas con papeles físicos. Como mi amigo David diría, tienenustedes que tener empapelado un par de cuartos con fotocopias de mi DNI.No puede ser que se dedique el 20% del tiempo a burocracias que no sondiscriminantes, no aportan nada más que pérdida de tiempo: las haces, nadielas lee, y ya está, aceptan a todos. Para dar clases, un papelito. Irte deviaje, otro, volver, otro, adelanto, otro, expediente compulsado, certificadopago tasas, certificado centro receptor, adscriptor, idoneidad, informe gastosprevistos, informe dinero en cuenta, de estar en el periodo de investigación deldoctorado, programa trabajo, factura original, otro papel, pasar a contratootro, empadronamiento para no sé qué, vida laboral, CV, sexenios directores,sus CVs, sus papers, su informe firmado por el director de turno, título

xiv Algunas críticas y sugerencias

licenciatura, idomas, cursos Doctorado, pago cursos, calificaciones, títulomáster, inscripción al proyecto, papers físicamente, papeles congresos, papelescharlas, memoria del grupo, otra burocracia, memoria del IFIC, otra vez aponer todo, actas, informes de no sé qué, que el vicerrector firme el 12 de cadames, que también el director del departamento, todos los papeles sellados aMadrid ... ahora todo ésto en inglés, español, valenciano y para un postdocen Italia en italiano... cojonudo...

¿¡De verdad es necesario todo esto!? ¿No es suficiente con que pida labeca entregando todo telemáticamente y si me la dan, ya está, me dejaninvestigar y dar clases? ¿Es necesario rellenar informes que no discriminan,que los aprueban a todo el mundo y solo me hacen perder el tiempo en lugarde investigar? Que es más difícil presentarte bien a Selectividad (día, hora,pegatinas, colores, no firmar, escribir por una cara, etc etc) que hacer bien elexamen, e ídem con la tesis... que David tuvo que hacer un pdf de 4 hojaspara aclararse con la burocracia... que le piden matrícula en el doctorado,inscripciones, informes de 6 personas (no sé cuántas mujeres, no sé cuántosde la universidad, no sé cuántos de fuera), y además de esto, ¡le piden hastala partida de nacimiento pare recoger el título! Por favor, que ha nacido...creedme... puede que la probabilidad de que nos haya engañado durante 30años no sea estrictamente nula, pero vamos a jugárnosla, va, ¡¡¡aunque sólosea por esta vez....!!! La última gran burocracia absurda, para un contrato,es pedirme un certificado de no disponer del título de Doctor, un título de notener el título, sería de risa si no fuera porque lo necesito de verdad, y a vercómo se consigue... Y de nuevo otra vez me piden el título de Licenciatura(cuando he hecho el máster, estoy inscrito en el Doctorado etc.), que les tendréque llevar tras descolgarlo de la pared, con marco y todo. En serio, intentenreducir la burocracia hasta el mínimo posible, por favor.

Y digo ésto yo que soy un mindundi, y ya constato como año a año lospapeleos que debo hacer crecen exponencialmente, no me puedo imaginarni quiero pensar lo que tiene que hacer un profesor, un catedrático, uninvestigador principal de proyecto, un director de departamento o el directorde un instituto. ¿De verdad hay que ahogar a la gente más brillante enpapeles?

Y un tercer canto al aire, aunque suene extraño: publicar demasiadodebería estar penalizado... casi debería haber un máximo de papers con losque alguien puede inundar el arxiv anualmente... Estamos inundados deinformación, crece exponencialmente, cuesta estar al día, y más aún separarla paja del trigo. Muchos papers no dicen objetivamente nada nuevo; nopodemos valorar la ciencia a peso, hay que encontrar otro baremo (tampocosólo las citas)...

Espero que alguien pueda hacer algo respecto de estos tres puntos, que

Algunas críticas y sugerencias xv

en mi modesta opinión son más que importantes... si no, al menos me hepodido desahogar. Y espero que en España se siga (o mejor dicho, se vuelva)a apostar por la ciencia, el conocimiento y la innovación como únicas víaspara el progreso y bienestar humanos. Fin.

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